How are we choosing the probability $p_{rr}$? Are there any choices of $p_{rr}$ that might make Russian Roulette unfavorable over just using N bounces?

Is $p_{rr}$ a fixed probability for all possible paths? By doing that, it would introduce some bias right?

I think the point of probability prr here is to cut off the infinite recursion. It is not a fixed value, should be determined from efficiency consideration. My understanding is that prr is small if brdf for the next bounce direction is low, but not sure if there is a mathematical relationship between the two.

How do we know this algorithm is guaranteed to terminate? Isn't it possible (though unlikely) our algorithm will keep rolling numbers under $p_{rr}$ and thus never return 0 for $X_{rr}$?