Not sure what you expect us to do. It's a physical issue so you're going to have to keep calling around until you get someone to come out and do it. Although be warned, the wait list for things like that can be quite long. I know we got a date that was 2 weeks out from installation.
"what modem & router will they probably supply?"
Probably not a router that's going to improve your Wi-Fi connection while gaming by much, if at all.
"Will the modem automatically be bridged or will I need to ask for that?"
Should always ask just to be on the safe side.
"Will their modem &...
If the trees are on your property, then you're the one responsible for trimming. Unless you're asking who to call to hire in order to deal with it, in which case you should look into lawn care services. You should find someone who'll be able to take care of it.
As for the cables, you'll have...