Cloelia, I enjoy reading your thoughts and am glad you won't drop the manga. I like how emotionally charged and analytical your writing is.
(Thank you, even if I'm always wrong about my theories *sighs*)
I only hoped it wasn't possible for Satan to possess Rin because Rin has his own demon heart that technically possesses him already. Apparently it is possible. Damn. Well now that I think about it, the fact that Rin's heart(demon power source) is blue flames that originally came from Satan, means it maybe can serve as an access point for Satan to possess.
I got a few more thoughts about this and what I said earlier.

It's not certain yet, but it really looks like Rin's demonic heart it's about to come out of his chest. Now, we talked over and over about Rin's demonic heart possibily been in Gehenna and all that. Now, staring at chapter 97 pic, makes me wonder if Rin's -demonic heart- was in fact inside of Rin (in Assiah) all along. But than I remembered about the thing of the sword and its acting like a gate. BUT what Amaimon said before (at least in the Viz's translations) is that Rin's POWER is in Gehenna. Amaimon never mentioned Rin's demonic heart. So the sword should serve to put in contact Rin with -his power-, not -his demonic heart-. If that's the case, then Rin's demonic heart was prolly really inside of Rin.
Then, there's this baloon speech I noticed while looking for the Amaimon's demonic heart panel. Mephisto uses these words: "Going overboard will destroy your host body. Then you couldn't stay in Assiah." So, I looked back at chapter 97: Rin's skin is starting to necrotize as his chest is sprouting flames (to release his demonic heart out of it(?)).
And I had this idea: the Artificial Gehenna Gate is still opening, there, in Russia. And this, as we said, was also part of Satan's plan in chapter 1: to open the gate, break the Koma sword and throw Rin inside of the gate. I'm not sure if I can add more thoughts about Mephisto's line since Amaimon and Rin are, essentially, two different cases. Amaimon got a host body, while Rin is "born" with his body. So I'm gonna skip this part.
But, out of all this, I got another theory. What if now that Rin's demonic heart should (I guess) separate from his body that would make Rin an "empty" shell? Maybe this is exactly what Satan is planning for. He wanted to break the sword to break the "seal" that ALSO kept Rin's heart inside of his body. Well, after all, as soon as the sword broke Rin didn't go berserk, but (after a few minutes) all he did was starting to necrotize and sprouting flames from his chest (and his body), so I guess if Rin's demonic heart will indeed come out from his chest, this should clearly be connected to the sword breaking.
And, once Rin's demonic heart will be out of him, Rin would be left "empty" BUT, after all the training with his flame power over the months, he'll be capable to sustain an even stronger flame power (Satan's) along with his (Satan's) demonic heart (if he got one?). So this is what Satan could mean by wanting to make Rin a vessel for himself? This is why Mephisto worked hard to train Rin for?
Yukio, a grey area, seemed more likely to be possessed. It happened in the anime too.
Sometimes, Kato-sama picked events from the anime and switched roles around. Like when in the anime it's Rin asking Shiro about their parents and in the manga is Yukio, so I would even expect/wouldn't be surprised if Kato-sama chose to switch roles in this case too.
He might appreciate Yukio's cold demonic brutality and cunning we all know he has. The unknown factor is Yukio's strange direct link with Satan that's not quite a possession.
Rin looks like he could really use the elixir now, but I think maybe his body can barely manage to adjust without it with more severe training. Yukio would need it more though I think. Plus he has a more direct access being in the Illuminati, and I think it's one of Lucifer's methods to help him get stronger: whether for Yukio's own use of flames to do Illuminati handiwork, or to adjust his body as a potential vessel for Satan as well. All good things for Lucifer. Yukio volunteering himself as a vessel as a substitute for Rin is a possibility, or Yukio just plain being easier to guide into that position is also a possibility.
Another thing is, if Lucifer gives Yukio what he gave Toudou and the other demon eaters, will that make Yukio a demon eater of Satan? That would be crazy and more cracky of a theory if say, Yukio had the potential to take over Satan completely. Well the difference between Yukio and the demon eaters is Yukio already has a link to Satan, while the others didn't and just hunted down the immortal demons of their choice. At the very least it would make him handle the flames better I think.
Honestly, I'm starting to think (I think too much, I know) this whole thing could be a lot more complicated. Based on what the manga stated/kept on stating/showed/proved/insisted on proving so far: Yukio's 100% human. He's not possessed, he's never been and even now HE's not possessed (unless, prolly just his eyes but there's no proof yet).
Yukio is the human son of a human (Yuri Egin (at least for as far as we know she was human. Clone or not)) and another (enhanced) human (Goro, subject 005). Yukio is 100% human. While, about Rin, my headcanon is that at the moment of conception, the same applied to him too. And, when the flame "gene" came into play, it changed things around (for Rin). Like, it "mixed up" and converted part (prolly 50% since he's considered "half" demon, but) of his human genetic heritage into a demonic one. Included his demonic physical traits like ears, teeth and tail (which were later "concealed" by the Koma sword seal alongiside his access to his flame power and/or even his demonic heart or something like that).
But, back to Yukio's case. I'm not sure if this can count as an evidence but I think if what Shiro stated it's true and Yukio (always based on Viz's translation) "couldn't BEAR it ('cause he was underweight)" maybe that means Yukio (he himself as a whole person) was born "unable" to bear a possession (and, I mean, not only for a physical matter (like he was underweight) which -I- consider something Shiro may have come up with to simplify his explanation to Rin). He just was born "unable" to bear any kind of possession.
To add to this, in chapter 86 Mephisto stated, when talking about the clones, this particular fact: "No matter how excellent the bodies were, they couldn't become
possession bodies." I never thought about this. Maybe the products of cloning experiments can get this kinda "side effect". The fact that "possession bodies" exist could mean there could also exist "NOT possession bodies". Bodies essentially UNABLE to be possessed. And I thought that this could be Yukio's case. I may be wrong about this part too 'cause maybe what Mephisto actually meant was: "They couldn't become possession bodies (for the Baal/strong enough to sustain a Baal for a lot of time)"
Anyway, what I mean is: my feeling is that Yukio could NEVER really be possessed by a demon. Satan or any other kind. Also, we wondered in the past with all the shet Yukio's gone throughout his life (the stress/the frustration/his fears and so on) it's plain weird no demon ever tried to possess him during his entire life. Shura herself stated a demon could take advantage of such things. But it never happened to Yukio. Aside his eyes. Which I already said what I think about (he may have inherited them from his human father and could be the ONLY part of his body actually able to bear possession ('cause all we've seen about that Goro guy so far are only his Satan-shaped eyes and etc etc), but I'm gonna wait on Lucifer for more infos (if he -hopefully- will finally feel like sharing some)).
So, in conclusion, I don't think Yukio himself (his whole body) could ever be used as a vessel for Satan (and I somehow think this is why Satan doesn't even give a crap about him and all that and he doesn't even consider Yukio as an "option" or "plan B vessel" (or even a son (well, if Yukio is 100% son of his human father, Satan's not wrong, after all)). Unless, the elixir could come into play and do something about this "problem" and turn Yukio's body into a "possession body".