You need to see if there's a ground inside the box
For your application, equipment safety ground MUST be separate from neutral. let's examine the cases.
There is neutral and a ground wire (or metal conduit)
The NEMA 10 does not connect to this ground.
You can extend off this outlet by tapping the two hots and ground (and neutral if your charger needs it)...
(I updated your use of 110/220 volts because that hasn't been widely available since the 1960s.)
The answer is no. NEC does not allow multiple outlets on a 240 volt circuit. Anyway, wouldn't you rather a 50 amp 240 volt circuit for charging your vehicle?
The floating neutral is good news, but you don't need a ground rod for this
The fact that your generator has a floating neutral is actually quite fortunate for you, as breaker-interlock-based transfer setups are largely incompatible with generators that have bonded neutrals (the more common case). However, since you're plugging your generator into your ...
Eventually, I decided to wire the outlet as a NEMA 14-50 (hot-hot-neutral-ground) rather than a NEMA 6-50 (hot-hot-ground) because the I can re-wire to a 6-50 easily enough by capping the neutral wire and from my limited research, I found that more EVSE chargers used NEMA 14-50 rather than NEMA 6-50.
As far as adapting the outlet to my current charger, ...