Conservation of energy means that the amount of energy, impulse does not change. In General relativity this might be a little different/more complicated. In that case conservation is true for flat space, but i am not sure about arbitrary spacetimes.
Thermodynamics itself is much more tricky, things in it really are not as undisputable as people claim it is.(just had a course about that)

The A->B's of your post is from the physics point of view, silly. There is not something as energy that is nothing else(like matter) just as there not something like impulse that is not something else.
Perhaps if people say that they mean radiation-like and matter-like, in which the latter refers to relativistic speeds. But mostly if someone says something like 'pure energy' he is just spouting out nonsense.

(2) I do not know exactly, have followed general relativity.(did not pass ) But i did not see a treatment of that, specifically. I could look at a solution for a expanding universe, but i am not sure it will answer anything. It will give aspects on the event, but not near t=0, since quantum has to be accounted too there, and we do not have sufficient theory of both. Maybe in a philosophy of science course i will learn more on that.

(3) Physicists might say stuff like that, but that is not how to argue, it is not sufficient to talk just about 'speeds of time', when space is curved like that. Also, to talk about anything in physics, one has to turn to the math.
Also other coordinate systems might not respect our ideas of time!

Note that time is more complex then how you are thinking about it, it is a dimension, just like the spatial ones. It is only because of the situation of space-time that this dimension 'behaves differently'. The origin of this different behavior stems from the(approximate) rotation symmetry of the universe, on the spatial coordinates. (probably mirror symmetries of spatial coordinates is enough for it, not sure)
Then there is the "arrow of time"; all the fundamental physical laws of nature are time-reversal invariant!! (and those that are not are phenomenological) Then why does a cube of ice melt, rather then that water mysteriously turns up cubes of ice! This is a mostly unresolved question of thermodynamics and statistical dynamics.

Edit: Note that i am just a physics student who hopes to get his bachelor this year.