I haven't done the math, but if we had significantly powerful lasers, shooting from the vacuum, isn't it very possible that we could actually deliver more acceleration to a far-away probe in 20 years than we could from Earth-based (or even orbit-based) lasers right now?

I have no idea what the best-case scenario for laser acceleration is.

No.

Let's say our initial boost got it up to 0.1c. After 20 years, it's gone 2 light years. If we make our space-based laser aperture really big, let's say 1km, then the light reaching our probe is something like a 25000km wide radius. That's not going to power anything.

If you slow down the initial 0.1c, then pretty quickly you're better off not sending it out at all and getting a bigger boost from your giant space laser being close for the initial acceleration.

The diffraction limit is annoying.