By the way, I'm new to proof of stake "mining".
Does it do any kind of brute-force search to mine blocks? I see stuff about a "target" and see it using a fair bit of CPU.
Yes. It's basically CPU mining, but the client will only try to generate a block when it is actually eligible. It is not mining 24/7.
(Still, would be interesting if the client was able to outsource mining to an ASIC as necessary. Wonder if this can be done, and whether it would give you any advantage?)