I'm thinking newbuild, albeit with some teething problems (unforseen at the end of TVH but that cropped up shortly afterwards) probably related to state-of-the-art computers and power systems freshly installed but not thoroughly tested, but Starfleet trusted that Kirk and Scotty and crew could work everything out. We don't know what the crew was up to between TVH and TFF, but I'm thinking the shore leave scenes at the beginning of TFF were well-earned after a few months of shakedown, working things out on the fly, and finally delivering Enterprise back to SpaceDock and taking a short break while the SpaceDock personnel began fixing everything...for once, there were no impending galactic crises and the regular crew could relax. The guys making loud noises on the bridge while Kirk was talking to Admiral Bob were probably SpaceDock crewers making scheduled repairs, not imagining that Enterprise was going to get called to duty abruptly in the midst of the work.
Mr Scott's Guide To The Enterprise says that the ship was originally named USS Ti-Ho, but I am thinking that he meant Taiho, named after a Japanese WW2-era aircraft carrier. IIRC, the registration number was supposed to be NCC-1798, also suggesting that this was likely one of the final Constitution-refit-class ships to be built.
If this new Enterprise was indeed formerly the Yorktown, that would explain some of the power glitches; aftereffects from the Whale Probe and such. But there is no suitable in-universe explanation of why they'd yank the Yorktown name off the hull and slap down Enterprise just for Jim Kirk, no matter how grateful the Federation President was. Also, IIRC there was a Yorktown as part of the original run of Constitution-class ships, the presence of Yorktown in TVH suggests that this ship was also refit from the original ship like Enterprise was, and hence would also be of around the same age, a spot over 40 years old, but with many years of service left in her.
Additionally, since Excelsior still had her NX-2000 registration in TSFS a short time before TVH, that suggests she was still in the prototype stage and it is unlikely that they'd decommission Yorktown, yank her name to reserve it for a yet-to-be built Excelsior (see below), then slap Enterprise's name on the old Yorktown. Nope, I'm thinking newbuild.
Regarding Excelsior, I am thinking that yes, she was still a prototype, and they wouldn't be building any more of her class unless and until they got all of her bugs worked out. The Transwarp Development Project history suggests it was quite some time until they gave up and put a "regular" warp drive on her. I am also wondering if perhaps NCC-1701-B was the second Excelsior built, using improvements learned from the long trial period of the NX-2000 (like sponsons on the secondary hull to hold more sensors or equipment plus additional impulse engines); subsequent Excelsiors improved upon the NCC-1701-B and fashioned the class into the "regular"-looking Excelsior we saw serving in TNG times.