it doesn't matter. put it on the axle or on the inside of the axle nut. either way, it's gonna do the same thing. I would try not to get any on the actual dropouts, though.

I agree with what else has been said, you probably just didn't tighten down your axle nuts all the way. Crank down on them hard!

Also, you don't want your chain too tight. You should be able to move it up and down at least 1/2" but preferrably more.