Can you direct link as an affiliate on AdWords search? Yes. What can you direct link? Physical product merchants (unless it's against their terms).
AdWords has a 1 unique display url rule. So, if two people are promoting the same offer direct linking, only one ad will be shown. Whichever ad is shown might have a higher bid, better quality score, or better account history.
I don't recommend using a tracker like P202 or CPV Lab for AdWords. I know some people use them, but I've seen some evidence they can help get you banned. Again, this is one of those things where you very well could run P202 just fine without issue. It's not something I do, or recommend though.
Normally, we would just place a pixel but this isn't easy to do on some of the recommended networks. What I do, is just track everything with sub id's. It's not ideal, but I've been running successfully for a long time like this.