Well, think I have some ideas about your questions - but only from a personal perspective - so plase take it for what it is!
On the snooping - although ever sinew in you wants to know whats going on, every time I have disovered something it has set me back days and possibly weeks. The thing is you only get half a story and your brain fights to fill in the pieces - usully the bad wrong ones. If you're looking my experience tells me that I rarely discover something positive. If you don't find anything your tendency will be not to think that things are not going on but that you're not looking hard enough! And discovery of your snooping will reinforce feelings in your W that you don't trust her. In view of the fact that you cannot influence W's actions and that you are committed to be in this for the long haul it makes no sense to snoop. Also it throws your focus back onto her and her actions. You need your focus to be on you right now. Do not be distracted from this by snooping.
I very much doubt if your W is laughing and joking with anyone behind your back. In all likelihood she is probably in a great deal of distress if not traumatised. She can't show you this though.She may be waiting to see how you react to see if there is anything in your actions which confirm for her action she should take. Don't complete the picture for her and give her the excuses she may be waiting for by reverting to type. If you want to keep your R with her and boys together this is going to be really tough but you can do it. What was it about you that she loved in the first place - what is it that you think she values in a R - write it down on paper and look for ways of mapping real actions onto those qualities. Don't look for big bang effects, just manageable, effective, repeatable changes.
Great dad, rubbish dad...same here - consider self to be a great dad when flying solo and completely rubbish when in presence of W - or at least did...not able to do right things, make right decisions etc. Its been a major, major source of frustration not to say upset and tear inducing as at times I have felt completely walked all over in my attempts to do the right things - not that there is necessarily a right or wrong, perhaps just different. Don't forget that yuour W is not likely to be able to see that there is a different way of doing stuff right now - just her way. I took advise from some folks here and aked her "I can see that you think I didn't handle that well, what do you think I should have done?"
Also I started taking my S6 away so he and I could spend some quality uninterrupted, non-undermined time together. If W sees that you can handle this and that your boys come back alive, happy and in love with Daddy it might go some way to changing perception. There's something quite powerful and empowering about a little boy constantly saying "me and Daddy...." I have found youth hostels to be great places for this sort of thing - they have other kids in them, they're well set up to make life easy - meals etc and there are often lots of kiddie friendly activities near by. Also look for something that you and sons can do that is yours.
Best - GFI
Me: 40ish W: 40ish Together: 20 ish years Married: 10ish Years