Yes - if you stop the cider before the yeast ferments out all the apple sugar, you can drink it in three weeks no problem. This is most easily done by using an ale yeast like S04 and cold crashing. Keep in mind that if you do this, it wont bottle carbonate - you will need to drink it still or keg carb.
If you let the sg drop below about 1.002, you will still have some good apple taste, but it will be dominated by the acids which will require weeks to months to mellow.
As far as the best SG for stopping it - its really a matter of what you like, as well as the acids present in the apples, type of yeast, etc
I would start tasting around 1.020, and when you think it tastes about right, crash it.
Commercial ciders are usually around 1.020, which is pretty sweet for my taste - but I suspect that is based on focus group testing that tells them this is what the majority of people like. My brewing friends tend to like it dry - down around the 1.004 - 1.006 range. My girlfriend and most of her friends like it sweet - that's why I stopped one batch at 1.026. I'd say the majority of my friends like it around 1.010, give or take a couple points, which is about where I like it.