Does the rotation speed of a synchronous machine affect the voltages induced?
The voltage induced is defined as the change in magnetic flux with respect to time (Faraday’s law). Which if you differentiate Faraday’s law you’ll find you that it’s directly proportional to angular velocity. Therefore increasing the frequency of rotation does increase the induced voltage. What really happens is that generators are locked at a specific rotational frequency, they always produce the same voltage, though not the same current. This is very important in order to not overpower the grid (if it’s being connected to the grid). The more power that is consumed the harder it is to rotate the generator, more torque is required. So if you intend to have the same voltage while under heavy use, you need to push the generator harder.

They produce sine waves because they are rotating at a constant angular velocity. The induced voltage changes as \$\cos(\theta)\$ (Faraday’s law).