Gnome BODY (important!)

I'd call that Stretching, Does Not Cross Intervening Space and Clairsentience, Only to Locations Stretched To. 
Main difference between my construct and dsatow's is that my version permits your detached bodyparts to be attacked to hurt you.  If that's not what you want, definitely use dsatow's version. 

Firstly, if I'm allowing a character to have "invulnerability" I'm not going to "well actually no" them.  I'm going to let them be actually invulnerable.  Because I told them "This construct is invulnerability" and I'm not going to lie to a player about mechanics. 
Secondly because I might allow invulnerability Only/Not Against THING.  Likewise I might allow LARGE NUMBER ATTACK Only Against THING.  And I'd expect both of these to work.  If Captain Invulnerable-To-Fire can be hurt by Ghost Burning Flame, something is horribly wrong. 
 
I am leaving this thread at this point because I do not feel this discussion is civil anymore. It is just two sides failing to engage with each other.  There is no point in anyone continuing in my eyes. 

But HERO System should be able to deal in Sith.  Or did Star HERO lie to me? 

As the old joke goes”sometimes the dice are out to kill yah!”

That’s funny cause we have some of the similar luck in our group too. 

We have a saying: "Aborting to dodge is useless."  Not because gaining a +3/+5 to your DCV is tactically bad, but with a lot of our luck, if you abort your next phase, the GM will end up always rolling less than an 8 making it effectively useless.  The last time I heard the phrase from my fellow gamers, they aborted to a martial dodge making it a 6- to hit and the GM rolled a 5.  To put icing on the cake, the GM rolled just over 4.5 points per die of damage.

"It depends".  One of the most effective tactics for a team up against a Big Bad (or even 2 against 1) is to wait for the opponent to commit to an attack, the target Dodges (or Blocks), the rest attack. 

One of first Fantasy HERO combats I did for the table I converted from D&D 5e was a practice sword fight (using great swords) between the estate guard captain and the party's Witcher character.
 
Once they realized they could have a proper sword fight (attack, block, counterstrike, block, attack, etc.) instead of just trading shots to the face D&D style they were really excited.
 
The players quickly adapted to HERO (minus power creation) and holding phases and Aborting for defensive maneuvers became very common.
 
I remember even back in the 90s my players when faced with a truly damaging opponent (like a frost giant say) would all hold their turns and then the character attacked would dodge/block and the others would attack like a pack of wild dogs.

My group had that too!  Very first time anyone Aborted they got hit through the Dodge two Phases in a row and KO'd.  First and only PC KO of the campaign at that, our resident KO-magnet wouldn't show up until next campaign. 

That's from the 5e version. 

With my more savvy and experienced players Aborting and Holding are big parts of the combat. The interaction between these and the SPD chart are one of the main reasons I like HERO combat.

Had the first session today and things went wonderfully!  Thank you again to everyone who helped me find this niche!  A very special thank-you as well to the individual who gifted me copies of Alien Wars and Terran Empire!  I wound up going with a Jhinu as a result and it's definitely informed some aspects of the character I wouldn't have thought to flesh out otherwise. 

Had the first session today and things went wonderfully!  Thank you again to everyone who helped me find this niche!  A very special thank-you as well to the individual who gifted me copies of Alien Wars and Terran Empire!  I wound up going with a Jhinu as a result and it's definitely informed some aspects of the character I wouldn't have thought to flesh out otherwise. 

I don't think it does anything that Trigger doesn't. 
Something like Trigger (+1) for ## AP of attack powers, Limited to only activate when a "minion" is defeated and only to target a "minion" adjacent to the defeated "minion" should do Takedown 1.  Trim off that last bit and add a Trigger for movement and you should get Takedown 2. 

Had the first session today and things went wonderfully!  Thank you again to everyone who helped me find this niche!  A very special thank-you as well to the individual who gifted me copies of Alien Wars and Terran Empire!  I wound up going with a Jhinu as a result and it's definitely informed some aspects of the character I wouldn't have thought to flesh out otherwise. 

To add to roles not covered, and to piggyback on "science officer," I am reminded that Star Trek, both original and TNG, often had guest stars be geologists, xenobiologists, botanists, and various other science specialists - you could be one of those, with a "side order" of other things/skills.
 
Likewise Cultural Anthropologists who evaluate a whole society rather than be "the face" who skill rolls a single NPC into an info dump.
 
Finally, I don't know whether the Terran Empire has Universal Translators as commonplace like in Star Trek, but one of my favorite sci-fi characters is Daniel from Stargate - linguist, sociopolitical analyst, archaeologist, etc.
 
The Galaxy might run on Galactic English as a lingua Franca, but I have to imagine that, were there no universal translator, many backwater worlds, and some economic powerhouses, would prefer their own languages.
 
(And yes I know that even now in 2020, Google and other companies are trying to have real-time audio translation)
 
So... Science specialist, linguist, anthropologist, etc... Some possibilities...

I don't think it does anything that Trigger doesn't. 
Something like Trigger (+1) for ## AP of attack powers, Limited to only activate when a "minion" is defeated and only to target a "minion" adjacent to the defeated "minion" should do Takedown 1.  Trim off that last bit and add a Trigger for movement and you should get Takedown 2. 

To me, the effect of desolid is "You and most non-you things do not interact", not anything directly to do with defense.  I think that's a lot of the reason for the disagreement.  If I approach desolid from a different angle, yeah, I can definitely see how it'd become the thing to use for "can't be hurt" effects. 
 
Here I completely agree! 
The +1/4 level always seemed silly to me.  If your SFX justifies your attack working against a particular SFX of desolid, why doesn't it just... work?  What makes this lightning bolt so different from the half-dozen other lightning bolts in your Multipower?  Why isn't the "weakness" in desolid letting either all the lightning or none of it through? 
The +1/2 level is even worse to me, to the point I actively wish it wasn't a thing.  What in the world justifies an attack working on ghosts and cloud-people and somebody vibrating through matter and somebody a quarter-inch to the fourth-dimensional "left" and people made of light, fire, and radiation? 

The problem is that, as Ragitsu mentions, attacks that fail due to desolid pass right through.  Even Cannot Pass Through Solid Objects doesn't help with this.  Which makes it pretty obvious what's going on, and is absolutely not the "Superman Effect". 

Correct me if I am incorrect, but...isn't Mr. "I'm technically invincible, but also quite intangible." going to make for a dreadful "human" shield?

I would not accept your "Always On but turns off" construction. 
I also wouldn't accept this SFX-mechanics combination.  Desolid is not for invulnerability, it's for inability to interact with and be interacted with by certain things. 
Furthermore, your proposed "Desolid, not against Mental Powers" is invalid.  Mental Powers already affect Desolid. 
I have no idea what you're trying to do with Continuous as a Naked Advantage in terms of RAW, I can't see any way that would do what you say it's doing. 
And even if your power worked, what GM would allow it? 
 
Even if a GM somehow accepts your construct, this offers absolutely no defense against surprise or invisible attacks because the Trigger is dependent on your senses. 
Likewise, generic power rules state you cannot both turn a power on and off in the same Phase.  Even if a GM accepts your construct, it won't activate if you've recently turned it off to be able to interact with the world around you. 
Even avoiding that detail, you're also rendered totally impotent (as you do note) by even the weakest constant damage effect. 
And lastly, by the definition of how Desolidification works, this can't provide some of the Life Supports you mention like protection from drowning. 
 
In summary, this doesn't work and even if it did it's nowhere near invulnerability.  There's much better uses for over a hundred points. 

By definition, 'invulnerable' means 'impossible to harm or damage'.  Per RAW on 6e1 p190, "Furthermore, a character must define the special effects of a reasonably common group of attacks that affect him while he's Desolidified." RAW then goes on to provide examples of what is meant by 'reasonably common group of attacks" using a few that have very clear cut special effects.
 
Put succinctly, Desolidification is NOT a defensive 'I win' button … and it certainly does not provide a defense against all attacks that meets the definition of 'invulnerable' given the above text from RAW … in addition to Mental Powers also applying.
 
I'm scratching my head trying to figure out the point of playing a character or NPC … in a game … that is 'impossible to harm or damage' (i.e. 'invulnerable'), anyway.  That just seems awfully boring, to me.  If the intent is for a GM-controlled plot device, then treat it like one and don't bother defining it, as GM fiat is fine. Then again, GM fiat is, IMHO, an unnecessary crutch typically wielded by weak-minded GMs .... that's also awfully boring.

This sounds to me like a conversation that should involve your group, not some random strangers on the internet. 
 
Tell them you are [enthusiastic/ambivalent/hesitant/concerned/whatever] about the idea but before moving forward to [saying no/saying yes] you want to have a dialog about [the questions and concerns you have, such as the ones you raised in your post] so you can be sure everyone's on the same page and onboard with the idea. 

Highly subjective opinion post below. 
 
The way I see it is that every power can be described both in terms of its flavor text and its mechanical construction.  The flavor text is the narrative description of what the power does and how, the mechanical construction is the rules description of what the power does and how.  These aren't highly distinct ways of describing something, like a painting and a photograph taken from opposite sides of the subject.  They're supposed to be nearly identical views of the same thing, like what my left and right eyes see. 
The flavor text and mechanical construction need to line up.  They're descriptions of the same thing, so they shouldn't be describing different things.  If I look at something and my left and right eyes are getting wildly different pictures, something is wrong.  I'm seeing two different things where there should be one thing.  I shouldn't see differences unless I get up really close and try to find them. 
If the flavor text is allowed to become disconnected from the mechanical construction, the narrative and mechanics have been ripped apart and the role-playing game is falling apart.  I, personally, am categorically incapable of role-playing in a situation where the narrative isn't describing what the mechanics are, I can play the mechanics game or I can role-play the narrative but I can't do both unless the two agree on what's going on.  That's why I wouldn't permit flavor text of "I turn into an elephant." without some mechanical construction that validates turning into an elephant. 
 
This is also why I hate "Magic" with no further qualifiers as SFX.  "This magic spell causes the caster to teleport, vanishing from where they are and appearing where they want to be.".  Is that the flavor text or the mechanical construction?  I can't tell, and that's a red flag.  It suggests to me there isn't any flavor text. 

6e1p279, emphasis mine.