Electrical – Do I need a separate switch leg for a light and a fan in a combo fixture

There is a major consideration that must be investigated before this can be answered in the positive. That is - Is the fan / light combo unit designed to support separate wiring feeds for the light and the fan. Many units do support this whilst some may not.

If your unit does support this then there is a second consideration. If the vanity light and the fan combo are on the same power feed from the main breaker panel then it is possible to entertain the idea that that the two lights can be interconnected to the same switches.

Since the two switches are in the same electrical box this is a good start if the prior considerations are met. The last criteria to look at is if there are already separate power feeds from the switch box to the fan/light combo in the ceiling. If not then it will be necessary to either replace the existing 2_wire+ground cable between the boxes with a 3_wire+ground cable OR to add an additional 2_wire+ground cable.

Once all that is in place the process of merging the two lights to one switch becomes one of pig tailing the correct sets of wires together in the switch box so the single switch can operate both of the lights.

Yes, if you have a 20A circuit for a 20A load, then you need a 20A switch. You can't put a 15A switch in there for the same reason you can't use 14 gauge wire - the conductor is to small, leading to a fire hazard.