Short answer
Don't formalize. Communicate more often.
Long answer
I think this situation calls for 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.
"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.
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: there are several ways to put everything down in time planning etc, but because people will tend to feel that a) things are now fixed in stone and b) there is no flexibility, this can become a tedious process. Better to call each other every day and change the schedule if needed (negotiate) than getting bogged down in contracting / formalizing.
To answer your last question: yes, you can put deadlines on the customer in the contract and/or time plan. However, instead of naming fixed dates, I would recommend to name your deliverables (update, patch, new release, etc) and indicate how much time is needed after the customer delivers their input/test results/etc. That way, when the customer is late with their results, they cannot expect you to deliver overnight.
Hope this is usefull. If anything incorrect / missing, let me know and i'll update my post.