Advice for living room upgrade

Last year, I was gifted a hand-me-down 100" pull-down screen and Epson Moviemate 72, that I set up for everyday use in my living room. The projector is 720p and 1200 lumens, while the living room has a large south-facing window (and a couple floor lamps we'd like to be able to leave on) and our cable is 1080p. So, time to upgrade!

We are going to upgrade the screen, likely to 120". Is a framed screen the way to go (and worth the extra cost for everyday TV watching)? Currently looking at this one and would love some feedback.

Most importantly, which projector?

What is the correct lumens level if we want to be able to watch during the day or with the floor lamps on, but also want the flexibility to have a movie-ish experience once in a while (lights off/low, at night)? Is between 2200 and 2800 the right range? Or a more specific window therein?

In addition to the size and brightness issues mentioned above, I would love to ceiling mount, flush with the opposite wall, which would be a throw of about 14', which appears to be longer than most short-throw projectors can go for 120" but shorter than some long-throwers. I'd prefer not to have to mount it in the middle of the room.

What's your suggested projector that meets all these specs? Budget is pretty flexible.

On that note, it appears the Epson 3000, 3100, and 3500 meet the above requirements (plus have lens shift), though they have a range of costs ($900, $1300, $1500). Other than more lumens than the 3000, what exactly am I getting for the higher cost of the other two?

Lastly, any suggested resources for directions on DIYing the hdmi cord(s) and the rear speaker wires behind/above the drywall?

TL/DR: Which 1080p projector and screen for 120", daylight/flexible, preferred 14' throw?

Thanks in advance!