These are simply my opinions based on some things i've learned in the past but...
(1) The universe is not infinite but it is ever changing.
(2) Matter is not infinite.
(3) Energy is also not infinite.
(4) Already answered...
If we agree that E=Mc^2 is true and we agree on the theory of conservation of matter, then we agree that matter can not be created or destroyed. However it can be changed. This theory has already been proven in the past in the "coal bed" experiment. Basically measured a liquid, the coal bed, and then streamed the liquid through the coal bed. After that the weight of the liquid was notably changed, however the difference in weight lost from the liquid, was equivalent to the gain of weight in the coal bed added to the "gas" or "vapor" that was captured from the coals as it escaped.
To clarify:
Coal bed (1) + Liquid (2) + Gas (0) = Coal bed (1.5) + Liquid (1) + Captured gas from coals (0.5)
1 + 2 + 0 = 1.5 + 1 + 0.5
3 = 3
Then their is E = Mc^2 which basically means Energy equals Mass multiplied by the speed of light squared.
Energy = (Mass) x (Speed of light)^2
Here is a documentary on it that i loved:
https://www.youtube.com/watch?v=YgpD4XZP0uM
This shows their IS a way to change mass into energy in some sense and vice-versa. Though the amount of energy to create mass is enormous, it is possible. Human's have used this to transmute other materials into things like gold. However the process costs more than the gold is worth and the gold also ends up radioactive. On the other hand, splitting matter a mass into energy releases enormous amounts of energy. The atomic bomb is the perfect example of such, or nuclear reactors if you prefer to be less destructive.
Anyway this leads me back to the original question: "Is the universe Finite?"
Well That depends what you consider the "universe." If you consider the universe to be only where their is "matter" than yes it is finite. If you consider it to be any spot where there is a black void than honestly I'm not sure, but If i had to guess I'd still say it's finite.
My view of the universe is as follows: The big bang occurred, energy was dispersed rapidly from the "center" of the universe. The heat and energy created yatta yatta yatta, created the planets and stars. You already get how that works. Some theorize the universe than expands infinitely...
but I'm not so sure abut that.
One thought i came to was "black holes." Basically a mass with a gravitational pull as strong as a sun, but so small that it is incredibly easy to get close to the center of that pull and become trapped. Many people imagine a black hole as a "hole in space" or a "Well" that things fall into but that's incorrect. It's actually more like a miniature sphere on it's own that pulls things in from ALL directions. Some say that black holes "move" some say "they pull things towards itself". Though motion is all relevant to the things around the object your observing... I'd argue it does both. *shrug*
Anyway, my theory is that black holes are the REVERSE of the BIG BANG.
Eventually i figure the black holes will collect everything in the universe and pull themselves together. Then with so much energy collected the black hole will become unstable and collapse. Causing another "big bang." Thus the cycle repeats continuously. No matter is lost, none is gained. Simply changed over and over into something new.
"I mean, yeah, the law of conservation of energy, but doesn't that mean we
won't run out? Which means it is...immortal?"
To clarify on this... if you mean that it never is used up, then your right to some degree.
Even after we breath in oxygen we still emit carbon dioxide. I'm not sure the exact ratios but the oxygen isn't simply destroyed. It's instead changed into carbon dioxide and our bodies use the oxygen and release some of it back out. Then the plants have at the CO2 and make Oxygen again. The cycle just repeats. Sure if their were no plants we would eventually "run out" of oxygen, but the original atoms that made up the oxygen would still be in the air, they would just be in Carbon dioxide instead of Oxygen [O2].