I'm going to tell you, as an A/V professional, what you have is typical builder grade. They certainly do not owe you an amplifier, and that's not a requirement for 7 cable drops, especially if they aren't all hooked up and hot. They don't know your cable signal, and not all amplifiers will work with digital cable systems, so you can't guarantee they will work at this point anyway.
I would say that the only request you've made which is at all reasonable is the one for them to properly label the cables. They should also be able to do so with proper wire labels, not with a Sharpie. That is something every company should do and do properly... But the rest? No.
As said, there is nothing wrong with terminating cat cables to RJ-45 jacks, as you may want to put them right into a switch, or into your router, or something else.
Yes, things are 'tight' in the box, but really - this is not their issue or requirement to change out. The builder highered a low-end company, and this is what you paid for. You didn't pay for a larger box, and you didn't request it ahead of time, Now ALL the wiring is in place and you are finally going to bring it us as a point of contention? I can't imagine a single professional A/V company who would not say "Yes, that would be nice to have what you are asking for, but is that what you PAID for?" If you have a contract which calls for a structured wiring cabinet and a dozen network drops and a half dozen cable drops, then that's exactly what you have, and is was professionally done.
If you now want a 42" space cabinet, I would say that you should pay for them to rewire, change your wall, and rewire everything as you expect. The work they've done is very nice, and I'm sure they will do an excellent job of it.
But do they OWE it to you?
Hell no, they don't owe it to you.
But, I think that you get that, and it looks like you will just tackle the work yourself which is a good thing. For what it's worth, I got an inexpensive label maker which allows me to wrap my wires instead of using tags which get caught up on things. This way you can label wires, but still move them around easily.