I also own the straightline
-The grease that came with it is just your garden-variety white lithium grease. Apply on all threads. It'll help protect them from wear and stuff apparently.
-I don't believe that squeegee type really makes a difference, as long as it gets it clean. I use rubber ones. One problem I noticed with the push-through squeegee's is that they get stuck on the inside of th stock muzzle breakwhen you're pushing them through.
-Yeah, I use hot water to clean all of my barrels, and then air dry unless you really want to try to get a towel inside of the barrel. Hot water gets all of the paint right off and won't harm your barrel.