I've recently replaced both battery and starter. Starter turns over real fast as it starts car. Will starter damage itself by being this fast or will it self adjust to a slower speed over time?
Short answer: Don't worry.
A starter is self-governed by it's windings. It will spin as fast as it was designed to and no faster. As the starter wears, it will slow down over time as the commutator and brushes break in.
It's very likely you just got used to having a weak starter and you're now experiencing how a proper starter operates.
Smaller motors and lighter oils will contribute to the starter seeming to spin a little faster. Especially when you put a new on on a car that certainly needed one. Inversely, motors that have high compression, that use gear reduction starters, will have their starter motors spin very fast, converting the high RPM's into torque via a gearbox.
Do not worry, it should not overspeed. If you are having issues with engagement (you can hear the starter spin freely without turning the motor if you are), ensure that it is properly aligned. The starter should engage without making any grinding/chattering/free spinning noise whatsoever.
Another thing to be wary of, is if you spend a lot of time turning over your motor, and it spins quickly, you may have low compression/worn piston rings. This can cause the whole motor to spin very fast during start up. If you're still concerned about how fast the motor spins during start up, I would recommend checking your compression, just to make sure it's in the safe ranges of OEM specifications.