No.

Huh?

The key word being assuming.

That's like asking: which came first, the chicken or the egg? I think the latest theory was that mass existed in a singularity and the Big Bang released that mass and energy. Which explains why the universe is expanding.

You're thinking about entropy here. google "heat death"

There is no before the universe, it just always was despite existing a finite amount of time. Well that's what the math says.. the rest is up to crazy philosophers.

If I'm not wrong, heat definition is derived from existence of matter(mass?)..

I'm no scientist At any rate, the stuff that was being converted was very hot, in the millions (or tens-hundreds of millions) of degrees.

Mass is a form of energy. I keep saying that!

Anyway, you can't transform all of the mass into other kinds of energy, because certain other quantities also need to be preserved (in particular, electric change and quark numbers).

You mean, when you sum all the contributions together, is the total net energy of the universe 0?
That's an interesting question. I'm not current on the latest cosmolgical models, but the idea is that, yes, it is. That means you need a large reservoir of negative potential energy (since the mass energy is obviously positive), which from what I remember people try to put in (the topology of) the event horizon of the universe. Not sure what the current status of that is or whether it's a particularly active field.
Alternatively, when you form a universe, it forms a pair, one with positive mass and one with negative mass.

No, that's more or less right. There is a (unsolved) problem with the idea though: you can only create particles together with their anti-particles. So the universe would need to be made up out of equal parts matter and anti-matter. Question: where is all the anti-matter? In principle there could be anti-matter galaxies out there and we'd never know it from looking at them. Doesn't really fit with how we now think large galaxies formed though (by merging lots of little galaxies), since you'd expect matter and anti-matter galaxies to merge every now and then, with lots of matter and anti-matter annialating. You'd be able to see that in X-rays (and we don't).
One solution is "symmitry breaking": matter and anti-matter are not exact mirror images of eachother, at least at high energies. Again though, it's not clear how you'd do this and I'm not current on the latest developments here.

I think you'd actually expect a lot of the mass to go into compact objects: black holes, neutron stars and white dwarfs. Once those form it's very hard to break them up again, although it is true that black holes evaporate through Hawking radiation.

Nope, that's effectively the same thing I said.

We're way beyond astronomical timescales at that point though. That's how long it takes.

But gravity would eventually bring everything back together, no? As I learned in another thread, thanks to Evert, it is impossible for things to be so far apart that they do not attract. Therefore gravity should eventually pull all energy back to the "center".

No, the gravity will constantly slow them down, but if the objects are receding fast enough, they'll be under less gravitational pull (due to increased distance) than the amount of slowdown. It's called "escape velocity". For instance, the Earth has an escape velocity of 11.2 km/second, if you had nothing but the Earth and some small object at the edge of the universe, this object would fall to Earth after an incredibly long interval and impact the earth at 11.2 km/second. So if an object were propelled away from it at 12 km/second, it'd never slow to a standstill.

Point well taken. Well, goodbye Universe. We knew you well