High volume SSL/SSH traffic. Systems can be configured to fall back to insecure random number generator under load but it creates an attack vector where you intentionally DoS a system, then have more leverage to crack the encryption based on the weak(read predictable) random data the encryption is based on. The practicality of such an attack is beyond my knowledge.
If the application running on the server needs to generate a lot of random numbers, yes.
Only if they need to be cryptographically secure - a PRNG, occasionally seeded with "real" entropy, is sufficient for almost all uses.
Yes, it can be. In addition to a security issue making timing attacks more practical if an attacker knows they can deplete the entropy pool in a repeatable fashion.