This is an assumption based on our current level of science and technology and failure of imagination. There are many possible ways around it: baby universes, false vacuum decay, even possibly conversion of dark energy (which appears to be inexhaustible) to normal matter, to name a few.

That may reveal a weakness in the doomsday argument itself. For instance, did any of the first hundred bllion billion humans even think of the doomsday argument? If the doomsday argument is flawed, will many future humans think of it more than briefly from historical interest? The nature of the human considering the doomsday argument significantly affects the sample space. Future humans in a free-energy universe would immediately see the falsehood of the doomsday argument and so from our perspective wouldn't even be eligible for the sample space. It may be that the doomsday argument only has a chance of being seriously considered by a 20th/21st century human, which simply turns the question into "what is the probability that I am somewhere between the 100 billionth and 130 billionth human to ever live?" or whatever the appropriate bounds might be.

I was unable to follow your logic there... Are you saying that the Doomsday argument is wrong or that it is irrelevant or what?

The doomsday argument is correct if no other information besides the total number of humans who have lived before me and finite resources are assumed. With additional information our confidence should be increased beyond what the doomsday argument can provide, making the doomsday argument irrelevant in practice.