stall or grinding during ignition sometimes
 

So far this has happened twice out of my last ~4-5 startups. Both times happend after I parked at a store or other building - i.e., car was warm. When I turn the key my car revs around 600rpm then dies. Both times a second attempt at ignition made it grind like a mountain cat with a hairball. Both times a third/fourth attempt while applying lots of gas got it to start up after a few seconds of hesitation. Then it would idle fine, run normal, etc., etc.

For the 2 months that I've had the car it would sometimes startup a little slow, but it was still within 4-5 seconds so I didn't think much of it.

Any ideas?

Several possibilities:
High mileage, low compression, worn/leaky ingectors, the RX-7 ECU's program for extra fuel on startup.
Individually these are no problem, but together they add up, and tend to make the engine want to flood on startup.

Some owners install a fuel cut switch to air out the motor when cranking, before turning on the pump.

The starter gear noise..
1) The enging may still be quietly sputtering when you hit the starter the 2nd time.
2) The 20 year old starter may be about worn out.
3) The ring gear may have some bad teeth.

mileage: 107,000

injectors: just serviced. Cleaned, some parts replaced (wires, fuel hoses, etc.)

compression: maybe, though power seems okay... compared to two rx-7's I drove with 200,000 miles on them :P. One had a good engine though, probably replaced. And a little more power than my current one, come to think of it.

starter, ring gear: ? Sounds reasonable...

i have the same exact problem going on with my supra right now. mine takes about 6 turns to get it over. and half the time it barely gets started. what is most likely the problem on your car is the same on mine. i havent pulled my starter yet, but i am assuming the the gear teeth on my starter are broken, or worn very badly and it is not engaging/grinding. i am hoping this has not damaged my flywhell yet. your best bet would be to probably get a new starter, or rebuild yours if possible.