Can a relatively small subset of random numbers be permuted and reused and still guarantee good expected running time for an algorithm like quicksort?
The question you're asking deals with the topic of derandomization, and you're proposing a specific technique for derandomization, namely using pseudorandom number generators. There are other techniques suck as using k-wise independent distributions and the method of conditional expectations. The holy grail in the field is proving the conjecture P=BPP, which states (informally) that we can always get rid of randomness, though the resulting algorithm could be slower; more precisely, it states that if you have a randomized polytime algorithm for something, then there exists a deterministic polytime algorithm for the same problem.

In your particular case, you don't need random bits at all, since you can use the linear time median algorithm to guarantee a running time of $O(n\log n)$ for quicksort. The AKS deterministic primality test is likewise a specific derandomization of randomized primality testing. P=BPP, in contrast, gives a general derandomization technique that works for every (polytime) algorithm; ad hoc derandomizations are still meaningful, since they could be more efficient (like in the quicksort example).