Can a transistor amplify a motor's RPM?

No. There is not much that you can safely use to get a motor to operate faster or on higher voltage than it was designed, lest it will overheat and sieze.

Yes, within the design tolerance of the motor, a transistor or other semiconductor arrangement can be used to regulate speed and torque of the motor within its design limits.

Battlebots are known for pushing 12V motors to 36V to get higher performance out of them:

a) A robot battle usually lasts less than five minutes;
b) Any given motor is unlikely to be running for the full five minutes, allowing it to cool down again;
c) The "over-power" is usually modulated to reduced heat build up;
d) If you're spending $5000 on something that could be destroyed in 5 minutes, the motors would be considered disposable.

If on the other hand, you are dealing with an induction motor, its speed is strictly governed by mains frequency less its slip-speed under load. You can obtain "phase-shifters" that allow such motors (usually on machine shop equipment) to be varied in speed.

At mains voltage you will encounter two types of motor: "universal" which can be run on equivalent DC,or "induction" which can only be un on AC. The layman, the best way to tell them apart is noise. If it screams and whines like a blender, vacuum cleaner etc., it's probably "universal". If it's suspiciously quiet, like the washing machine, or dryer (apart from some vibration), then it's probably induction.

Unversal motors are fairly cheap to manufacture (all wire, no magnets) and can be speed controlled by interrupting the power.

Induction motors that require speed control (agitate or spin the washing), either have two separate motors, or some have combined windings to allow for speed/direction change in one unit. Think of an induction motor as a fixed-speed fixed frequency stepper motor (very crude, but it may help cross the bridge of understanding).

No, but it depends. If you mean the motor will go faster then the RPMs it is rated for then no unless you put even more power into it but the motor will burn out if you put too much in. The transistor can however be used to control the RPMs but will not make them go higher then if you just plugged it into your power source.

If it's an AC motor you can use a circuit which involves transistors to vary the AC frequency, and hence rpm. If its' a DC motor you can use a transistor to control the flow of current and hence rpm. Which motor do you have / want to use? L

Um, depends on the motor. Some clock motors depend on 60 cycle (or 50, depending on country) to keep time, so changing the power would change the speed of the motor. If you have a motor with brushes, you MIGHT be able to time the impulses to hit only when the brush was in contact with the rotor, giving a power boost, but you would have to keep them in perfect sync or you would have pulses of speed or not work at all. Why do you think it would work?