I see several things going on here that need to be addressed seperately, but they all come down to a single thing: better expectation management (communication). Make explicit what you need from the customer and when, and make explicit when they can expect something back and what.

1) "emergency" fixes needed overnight:
If you mean that the moment the customer finds bugs, they need to be fixed "right now": is it fair for a customer to expect this? I think not. Bugs take time to fix and the team should get that time. Tell them beforehand, that they can expect their fixes "X" days after they report them. If they object, negotiate. In any case, make it explicit when they can expect their fixes.

2) nothing for a week:
So what's going on here? Are they not using the product or just not talking about it? Radio silence? Let me know and i'll update my reply.

About the formalizing: here be dragons. There are several ways to put everything down in time planning etc, but because people feel things are now fixed in stone and there is no flexibility, this can become a tedious proces. Better to call each other every day and change the schedule if needed (negotiate) then getting bogged down in contracting / formalizing. 

Hope this is usefull. If anything incorrect / missing, let me know and i'll update my post.