Hello...
I am about to install a new circuit in my basement to provide receptacles for small appliances such as a TV and a sewing machine. I understand that GFCI protection is required for "unfinished" spaces.

1. Exactly how is "finished" defined for this application? The walls and floor are painted, part of the area is paneled. The ceiling is not finished - it's just the exposed floor joists. There is a workshop, laundry room, furnace room, and some exercise equipment in the basement. Is this considered "finished"?

2. If I need GFCI protection, I need to decide whether to try to protect the circuit with a receptacles or install a GFCI circuit breaker. About how many receptacles can a in-line GFCI receptical protect without getting a lot of phantom interrupts?

Unfinished or not you need GFCI just because of the washer. You need at least two circuits in the basement, one for lighting and one for appliances. You should try to give us the wattage for all of your appliances, you may need a 3rd circuit. The GFCI could be a breaker or it could be a GFCI receptacle at the head of the circuit. The rest of the outlets down stream can then pick up GFCI protection, follow the directions in the GFCI box.

GFI protection is required for any outlet in an unfinished space of a basement, except, outlets for stationary appliances, such as washers, dryers, refrigerators, sump pumps, etc do not need GFI protection.

Having the walls painted and no finished ceiling would not constitute a finished basement.