I just completed some testing on double shot and thought I'd share the results with anyone that might want them. Firstly, proc rate appears to be 40%. In my testing double shot proc'd 82 out of 212 times which is 38.68%, due to the sample size it's probably safe to say it's an even 40%. Secondly, both shots are completely independent of each other. Both or either can crit, miss, and deal separate amounts of damage. Double shot can proc on missed attacks. You also receive full tp for any additional shots from double shot. Double shot does however share a recast timer with unlimited shot, and unlimited shot merits do not effect the recast timer of double shot (This I heard second hand).
I've also done some testing with refulgent arrow and double shot. Average damage overall at 100% tp seems to be approximately the same but there are occasional damage spikes with double shot active. It seems unlikely that it has any effect on weaponskills, if it does it certainly has a lot lower proc rate than with normal attacks. Some additional testing with higher tp might be required