Is Primecoin mining competitive? Meaning, if more people mine Primecoin, does it get harder to mine? From what I understand, it shouldn't, but I want to be sure.
Primecoin mining is doubly competitive (in the sense you described). In fact, the amount you will mine with given power is inversely proportional to the cube of the total network compute power.

As in Bitcoin, a difficulty parameter adjusts based on the mining rate to achieve a desired block frequency. Not only that, but the mint rate is inversely proportional to the (square of the) difficulty. Unlike Bitcoin which has a fixed schedule, which at worst can be slightly accelerated with increasing mining power, in Primecoin the total number of coins that will ever be generated will decrease with increasing competition.