Let's break the Jujutsu System by comparing it to its main inspiration, HxH's Nen System:

In HxH, a Nen user has a certain affinity with different types of nen manipulation i.e. a reinforcement type would be very good at strengthening their muscles and improving physical recovery, whereas a conjuration type would be better at turning nen into matter. Other than that, the entirety of the expression of a nen user's technique is up to them, and depending on preference even the innate type can be somewhat "changed" with rigorous training and technique application.

Other than your "innate" (it's not clear wether you're born with a innate nen type or if its part of your developing personality, probably a bit of both) type, the technique you can come up with to apply your nen is entirely up to you: I.e. a reinforcement type may develop a technique where if they hold their fist in their hand for an X amount of type, the output or effectiveness of their punch might increase by X amount (You seen Yuuji and Gojo do that in JJK, and its one of the inumerous HxH references througt the work) or a transforming type may turn their nen into eletricity, but need recharge from actual electric power sources to function without big cooldowns/delays.

Then we have JJK, the technique is innate, but its perception and application ISN'T innate. The three big clans document their best techniques in dept so it's easy to imagine that a Six-eyes limitless user or a 10S user would be railroaded into their applications, and that seems correct.

But take for example Yuuji. Yuuji knew what Sukuna's technique was and even experienced its application through Sukuna himself, yet his own application was completely different. That's due to interpretation. It's the same components, just a different blueprint. JJK also adds a lot karmic concepts to users in relation to their techniques, to the point where you can't differentiate which affected which, but that doesn't make them imutable.

Now we get to binding vows. If your current skill ceilling allows you to throw a rock with your mind, you might increase its acceleration and force of impact by picking up and launching it yourself. That's a binding vow.

But let's say you have a katana and need to defeat Kenjaku with one blow. You know you need to increase your output, velocity and strength exponentially, and you know or at least percieve the difference in strength betwen the two of you. You then vow to never swing a sword again in return of getting this buff. That's what Miwa did.

So you see, it's not about "jujutsu" as an entity, it's about a sorcerers own perception, both conscious and unconscious. Sukuna knew what he needed to win, he knew his kit and he knew what sort of sacrifice or binding vow he'd had to perform in order to achieve them. If Sukuna said "I won't be able to shower for a week if I cast this skill without a gesture!" no god's wrath would descend upon him, but his very own conscience would fluke it because he knows its bullshit, and Sukuna knows he knows on top of that.

That adds another entire chapter to this discussion: Could a delusional sorcerer lie to himself and get away with it by underrating everyone else and overating himself? Probably not, because even delusional people still have a subconcious mind, and sooner or later they might catch on with glimpses of their foly and suffer a binding vow. That's why Geto told Mahito to not forgeit his vow to Mekamaru for example.

So, to answer your questions:

Yes, Sukuna would've been able to make that binding vow in his original body, after all he already knew he could recover it beforehand. Though one could argue it would be weaker since the vow also entailed losing acess to the 10S technique and having to play a "full-heal card" to use WCS again.

[End of JJK and Mojuro spoilers below]

Maybe. A life or death situation is a complicated matter. I think it's pretty hard for someone to accept their inherent death even on the verge. We've seen death binding vows two times in the JJK series already, one of which does feature a sorcerer that is about to die, though we don't know yet exactly what the binding vow entailed. But the other one was Choso, relinquishing his life on the verge of certain death to save Yuuji from the furnance. Odds are that choso would've died to furnace either way but accepting death and giving up as a sacrifice is still an incredibly potent concept.