I would not recommend it, for the obvious reasons of dust and noise. I've had basement wood shops for my entire life and this fall will be the first time (I'm 46) that I will have a dedicated outbuilding for woodworking.
So - spare room. Dust is #1. An air cleaner and a dust collector are pretty critical. A spare room might actually work out better than a basement shop. Buy an air cleaner or make one out of some HEPA filters and a cheap box fan. You can open a window and direct the box fan outdoors to exhaust air. Noise won't be a problem if your family is OK with it. Another idea I saw recently is to replace the door to the room with an exterior door. The exterior door will be insulated (therefore more soundproof) and also be weatherstripped all the way around to help control dust.
Also, consider doing a lot of hand-tool woodworking. Not dust-free, but certainly less than with power tools, and most of it won't be airborne particles.
I'll second the advice on a hard floor surface. Laminate or hardwood would be a good floor choice. Alternately, you could remove the existing floor covering and just lay down a "temporary" floor out of T&G plywood or OSB, so that you won't have to worry about damaging the floor.
Lots of projects to do in a small shop. I find that I am never looking for projects, they come to me and I build them. It's called a "honey-do" list. If you are really looking for stuff, find out what interests you. Woodcarving is great for a small shop. I particularly enjoy turning, and you can set up an amazing shop in a small area - a lathe and a good bandsaw, plus some hand tools, are about all you need. Boxmaking, marquetry, musical instruments, the list is endless. I'd simply try to avoid projects that are too big for the space you're in.