If you snoop,make sure you use the data only as reference and not use it to get back to H or get mad at him. Only so you know if contact is going on. Having said that, it is EXTREMELY DIFFICULT not to be affected by snooping.
"Some people say that since we are no longer married, he can talk to who he wants. I don't buy that. Sure he can, but I just won't be around. " I think YES, it is his choice to talk to whoever he wants. It is his choice to lie or be truthful to you about the conversation/contact. If he lies, it is your choice to not be aruond and not include him in your life. If he is truthful, then you can build up the trust from there. If he even go further than that to be more committed, then I'd say you are moving forward. So I think you are right, he can, but YOU have the choice of not being there for him. This is tough for you. Take care.