In answer to your first question, in a typical PLL made from discrete parts the phase detector is followed by a loop filter. If the loop filter contains an integrator, then the loop will stabilize at a phase such that the average output of the phase comparator is zero.

I can't speak to integrated circuit practice -- having an integrator in the loop filter would mean either an internal (hence BIG) capacitor, or an external capacitor with it's extra connection and circuit complications.

In answer to your second question, I suggest that you write out the expression for a sine wave multiplied by a square wave at the same frequency and at some arbitrary time offset, and calculate its average value. You should see that it has a DC component that is equal to the sine of that time offset.

In answer to your first question, in a typical PLL made from discrete parts the phase detector is followed by a loop filter. If the loop filter contains an integrator, then the loop will stabilize at a phase such that the average output of the phase comparator is zero.

I can't speak to integrated circuit practice -- having an integrator in the loop filter would mean either an internal (hence BIG) capacitor, or an external capacitor with its extra connection and circuit complications.

In answer to your second question, I suggest that you write out the expression for a sine wave multiplied by a square wave at the same frequency and at some arbitrary time offset, and calculate its average value. You should see that it has a DC component that is equal to the sine of that time offset.