So this might be a dumb question, but do all 3 servers start new rounds whenever a share is found in any one of them? Or is each server basically acting like its own pool, each having its own rounds?

I ask because that will affect average time for a round to finish. Obviously if all 3 are pooled togehter, you'll find blocks 3x faster, albeit with lower payouts.

Rewards are combined between the 3 pool servers.  This is the exact same as running on a single large pool, I've just split it across multiple datacenters in order to provide members with the option to manually point to a closer server.