It needs to spin in both directions/back & forth, but only about 90 degrees & very slowly.
How slowly? One revolution per second? One per minute? One per day? One per week?
When it stops, how critical is the position?
The turn table needs to hold about 100 or so pounds, so I'm thinking I'll need a motor with a lot of torque.
Not necessarily. Good bearings under the platform would reduce the amount of torque required, as does the 18" radius there the load is applied.
I'd like to put a motion detector in the Arduino and when it sees motion, rotate a few times, then wait for the next time.
When the degree of rotation is limited to 90 degrees, how can it "rotate a few times"?
but don't know what motor & batteries I'll need.
You haven't given us enough information to help.
Do I need a motor shield or an ESC?
Depends on the kind of motor you end up with. For a servo, no for either. For a stepper motor, yes for the shield, no for the ESC. For other motors, yes for the shield, maybe for the ESC.
Or what type of gearing I'd need & if I need brushes or not.
Can't answer about the gearing without knowing the speed and the motor type. If you are painting, brushes are a good idea.
Maybe I could use 1 or 2 12v batteries....
Pick the motor, first. Then, get batteries suitable (voltage- and current-wise) for the motor (and Arduino).
I'm also wondering if you can ave a shield on top of a shield?
Depends on the two (or more) shields.
If possible, I'd also like to have sound with the wave shield & maybe blink a few LEDS...
Can I do all this with one Arduino?
Is "a few" 10 or less or 1000 or more? The devil is in the details, as they say.
Most likely, a single Arduino will be plenty.