Don’t dick around with infinite sequences and limits. One third of the faces say “reroll”. You stop only when you hit a face that does not say “reroll” and those faces are evenly divided between “resuscitate” and “don’t”. That’s a much easier way to show that it’s 50-50. 
The d3 is either a prism with pointy ends like Giles described, or a relabeled d6 (with two sides each labeled 1, 2, and 3). The d5 might be a relabeled d10 or d20, but is probably a triangular prism with flat ends, and isn’t necessarily fair. Likewise the d7 is probably a pentagonal prism. The d14 is probably a skew dipyramid (AKA a dual antiprism), like the d10, and is fair but boring. The d30 is a standard (though rare) die shape, is completely fair, and has the same symmetry as a d20 or d12. And the d100 looks like a golf ball and isn’t particularly fair, but is close enough that most people don’t care.
It will. If we define the probability that you roll a “live” to be p, then conditioning on the first step gives us that p = 7/20 + 6/20*p. You solve that, and you get p = 1/2.
I’m not sure what you’re saying here. There’s clearly some chance that you will reroll at least once, and the patient dies before rolling a live or a die. (actually, you do definitely roll a die each time, but you know what I mean).
Footnote to my earlier post - it is not necessary that one-third of the faces say “reroll”, only that all of the others are evenly divided. The preponderance of rerolls only determines how long the whole silly process takes. 
I have no idea what “then conditioning on the first step gives us that” means. Did you drop a word?
In any case, I think you’re forgetting that “roll again” is the same thing as “death” when it’s your last roll before time runs out. So death is actually p(D)+p(L)*(1-p), where p(D) is the probability of dying before the last roll, p(L) is the probability of getting to the last possible roll and p is the probability of rolling “live” on a single roll.
Or maybe they’ll resuscitate you just to ask whether you meant a 20 sided die or a 21 sided die.
Oh, but what happens in the (arguably fairly unlikely) event that you are still rolling “rerolls” by the time the Sun goes Red Giant all over Earth?
It’s all very simple:
(1) Roll the dice;
(2) After one minute, re-roll the dice (if necessary);
(3) After half a minute, re-roll the dice;
(4) After a quarter of a minute, re-roll the dice;
and so on, reducing the time by a half each time.
By my calculations, you will have rolled the dice an infinite number of times by the second minute, in the unlikely event that every result of rolling is “re-roll”, so won’t have had enough time to die.
Exactly. Just flip a coin.