1. This house doesn't have separate vents (a "wet vent" system?) - can I keep that in place, or must I convert to dry vents?
Each fixture needs to be vented separately, but the vents can come together above 42" high in the wall. A pro can define/describe this better.
2. Must each fixture have a separate drain connection to the stack, or can they connect to a large branch line?
They can connect together to a large branch line. Each fixture is given a rating in Drain Fixture Units (DFU) which corresponds to it's drainage flow. A pipe of X diameter can carry X drain fixture units. (Less if horizontal, more if vertical, usually) These vary a bit by code, but it's a way to determine what branch sizes you'd need.
For example, according to the code my state uses, it has a chart listing a Bathtub at 2.0 DFU, a sink at 1 DFU. A 1 1/2" horizontal pipe can handle 3 DFU in our code, so I could run both the sink and tub into one horizontal branch, but nothing more.
3. The run from the closet flange to the stack is perpendicular to the joists. Can I have a vertical drop from the flange to an elbow to avoid cutting/notching (I have plenty of space to work below the floor)?
As long as you are able to maintain proper slope through any horizontal run to the stack, you can drop below the joists to avoid going through them.