I’ve read that the Universe is 13.7 Billion years old. I *seem *to remember that also It is flat, accelerating, and will continue to do so. But for how long? (Yes, I’ve been watching Carl Sagan’s programs.) What’s the latest cosmological skinny on this? Heat death or Cold? And when? (I need warning so I can pack my suitcase) 
They’re still gathering data.
And I don’t think it’s flat.
There are three possibilities for the Universe. Which it is depends on the total mass of the Universe and its density.
- Open (will expand forever)
- Closed (will expand then contract, aka Big Bang/Big Crunch)
- Flat (will slow down in its expansion till it holds a steady size…neither expanding or contracting)
The latest data from the WMAP satellite shows the Universe is not only still expanding but is accelerating in that expansion.
So, the Universe is Open near as we can tell today.
I don’t think you’ll find many astronomers who still think the universe might collapse into a Big Crunch.
ETA: This means Cold, definitely.
Those were what everyone thought were the three possibilities, about 15 years ago (with the nitpick that the marginal case continues expanding to an unlimited size, but with a speed approaching zero). The thing is, though, in all three of those models, the Universe should be decelerating (at different rates in the different models, but decelerating in all of them). The actual observation, though, is that the Universe is actually accelerating, speeding up in its expansion. Clearly, there’s something that was missing in the previous models. We call that something that was missing the “cosmological constant”, or “dark energy”, but we actually know very little about it (aside from the fact that it apparently makes up about 70% of all the “stuff” in the Universe). Note that the existence or not of the dark energy is independent of whether the Universe is flat or not: One could have a positive-curvature universe with dark energy (which may or may not eventually recollapse, depending on how strong the dark energy is versus how strong the curvature is), a negative-curvature universe with dark energy, or a flat universe with dark energy (both of which will expand forever). Our Universe appears to be flat with dark energy.
Now, then, to answer the other question in the OP, this state of affairs will continue for as long as the dark energy continues in the way it’s going now. How long this will be, we can’t really say, since we know so little about what the dark energy is. The Universe once contained something qualitatively similar to the dark energy, but much, much stronger, in what’s called the inflationary epoch, but that eventually ended, so for all we know, the dark energy we’ve got now might eventually end, too. Alternately, it might actually get stronger with time, which could lead to an end of the Universe called the “Big Rip”, which would expand so fast that it would tear everything apart, right down to subatomic particles. The best data we have is easily consistent with the assumption that the strength of the dark energy is constant, which would be the simplest case, but we really don’t know.
Or there’s a really big, massive ring just beyond the edges of the universe such that the edges are tending towards it. So, maybe it’ll be a cosmic splat instead.
Of course, we don’t know what it’s doing since we don’t know what lay beyond how far we can see. No light, no data, no workable theory.
How can it be flat with Dark Energy if expansion is accelerating?
Sounds pretty open to me.
“Flat” does not directly refer to the kinematics of the Universe, but to its geometry. There’s a nice simple relationship between the two in a universe without dark energy, but the presence of the dark energy throws it off.
Basically, if you construct a really big triangle in a flat universe and measure its angles, they’ll add up to 180 degrees. If you do the same in a positive-curvature universe, they’ll add up to more than 180 degrees, and if you do it in a negative-curvature universe, they’ll add up to less than 180 degrees. In our Universe, it appears that the angles add up to 180 no matter how big the triangle is, so we say that our Universe appears to be flat.