Yeah, there is player advantage. Generally, green openings are good for P2, blue openings are good for P1. Red depends. Vivid drone is super good for P2. Wild drone is pretty good for P2. The red unit that makes green stuff is very good for P2. Odin is extremely good for P1 (if you want him right away).
Overall, my feeling says that P2 is a fair bit better on average (which seems to be backed up by stats, although I hadn't heard of them before).
Somewhat unrelated, I think the "first" and "second" terms are confusing. You could just as well make the first player have second turn but let him start with 8 drones. It's not that one goes first and the other one second, it's just that you play... offset?
Generally, you should have the better position at the board whenever you finished a turn, and the worse position whenever you start a turn. P1, you start with 6 vs 7 drones, you end with 8 vs 7. The farther the game goes, the bigger this gap gets. If, say, your opponent has an even position with you at the end of his turn, that means you're pretty far ahead.
By the way, Doomed Drone makes everything better. If I could decide, I'd make it a base unit.