I am using a 5 volt stepper motor from adafruit, the 28BYJ-48,
That motor has a winding resistance of at least 40-50ohm, that means it will draw a current of max 188mA at 7.5volts
At 5 volts (not coming from the arduino BTW, external supply) It cannot spin the disk that is required. At 6 volts, it is inconsistant. At 7.5 volts, it works like a charm.
My question however is this; Will I damage the motor by running it at this speed for extended periods of time?
well, you can let it spin at desired load and speed and check it if overheats and how fast. I guess you can use the sensors in your fingers for a balpark initial estimate.
The motors are damaged by overheating because high currents and/or limited cooling, long before a higher-than-rated voltage will do other kind of damage.
Also, I am using a 1A H-Bridge, but the power supply is 2A. Will this damage the H bridge, or does it regulate the current?
No way, the power supply rated current is the max it can give. It's normal (and required) to be higher than the load will draw. A car battery can provide hundreds of amps, that doesnt mean it will burn an arduino board that draws 100mA. If it is powered through the right pins of course.
The current is limited by the motor's resistance, and as I mentioned above, it is high enough to keep both the H-bridge and power supply within safe limits.
All you need is to let it run and check if it overheats.
PS as an assuring exercise
- measure the winding resistances of the motor coils at its terminals. Doesnt matter wich wires, measure them all and see what values you get
- measure the mA drawn by the H-bridge when the motor is spinning, then hold it with your hand, see if current changes.
- measure the voltage that motor windings see from the H-bridge, see if it is still 7.5volts or less. Some bridges have a voltage drop themselves.