Peeta is basically dying by the time Katniss finds him and the medicine that's in district 12's backpack at the feast was totally necessary to keep him from dying. The medicine is the backpack is in the form of a syringe and it's really advanced/expensive, way outside the realm of a parachuted sponsor gift, although they have a lot of sponsors and Haymitch sends Katniss lots of gifts, including their favorite meal (on silverware!) when Katniss acts lovey dovey enough. There are no notes with the parachutes, Haymitch communicates solely by sending or withholding the gifts. Haymitch does send sleeping syrup to Katniss- she feeds it to Peeta to knock him out so she can go to the feast. Presumably, Peeta was going to stop her from going otherwise and he's mad when he does wake up.

However, the medical care in the capital is super duper advanced so that as long as you're still somewhat alive by the time the doctors get at you you'll be good as new, which is why Katniss doesn't give up hope. [Also why Katniss made the decision to eat the berries so quickly: Peeta was basically saying that he's going to die soon anyway and he may have ripped open his wound (from the fight with Cato) and asked Katniss to shoot him] Yes, Peeta does lose his leg and he gets a prosthetic one as part of the medical care. However, it's super advanced so it's totally functional.

I think it's a good call to leave out Peeta's prosthetic leg for the movies. It wouldn't add anything and it would be complicated to do. They could have had Peeta being more injured at the end but even so without making it clear that he would lose his leg. They left out other stuff that had a bigger part in the book- like when Katniss' dying from dehydration in the beginning when she couldn't find water for the first couple of days.

[The dehydration is the first example of Haymitch communicating to Katniss: when Katniss is at the point that she thinks she's absolutely going to die of dehydration, she whispers "water" and waits for a parachute. When the parachute does not come, she figures out that it must be because Haymitch knows that she's actually close to water, so then she keeps going and finds water. This is at some point on the second day and given her extreme activity level, it's in line with when a human being would be at a critical level of dehydration]