Does the Hubble constant only apply to the current motions?

Yes.

Would it change with time?

Yes. And no. Hubble's Parameter changes through time. Hubble's Constant is what it is at the moment - so it will always be what it is, given what time it is. Which does make 'constant' a bit of an odd term, but physics nomenclature is bit of a slave to tradition

How would you prove either constant constant or variable with age?

It's derived from General Relativity via the Friedmann equations. So that's the mathematical derivation / proof. Empirical proof comes from more and more precise measurements and clever inferences.

Why does velocity vary with distance? Is this where dark force/matter come in?

Good question and I'm not sure there is a definitive answer but the general gist goes:

There is something intrinsic to space that causes it to expand. So think of a unit sphere of space - it expands to twice the diameter. Now there is more space. And thus more 'expansion stuff'. From the centre of the sphere to edge there is twice as much stuff expanding as there was before so the edge is now being 'pushed' away at a greater rate than stuff only one unit away from the centre. The rate 'per unit' of space stays the same, but the more units of space there between point A and point B, the more space is expanding so it accumulates over distance.

That's why more distance = faster expansion between those two points.

The fact that the equation comes out the way it does, and the observations leads us to the conclusion that there is something intrinsic to space that causes the expansion. What that intrinsic something is is not completely understood I believe, but yes - a constant energy density to space - ala a cosmological constant with Dark Energy being the culprit of this is a commonly accepted understanding, though there are some weird quantum field ideas that I don't understand that have been proposed too.