Lots of questions, so I will try to address them.

1. Throw distance of 11' is irrelevant without including screen size. The AX200 is a VERY versatile projector. It includes lens shift and a long zoom range. So, the AX200 with EXACTLY 11' lens to screen (throw distance), can produce an image between 55" and 110" diagonal.
https://www.projectorcentral.com/Panasonic-PT-AX200U-projection-calculator-pro.htm

The BenQ TK700, from the exact same 11' throw distance can only give you an image size from 104" to 134" diagonal.
https://www.projectorcentral.com/BenQ-TK700-projection-calculator-pro.htm#calc

If you have a 100" diagonal screen, then the projector lens would need to be several inches closer to get the size you have.

Two takeaways - Be EXACT with your throw distance. A few inches difference can matter a lot. And, what is your screen diagonal size, you can't do anything without knowing that as well.

2. Your projector mount, while a pretty cheap model (I do not like them), should still be plenty sturdy for any typical projector up to about 25 pounds in weight or so. It is a 'universal' mount, so the legs on it can be adjusted for any typical projector like those from BenQ, Epson, Optoma, and others. One thing I can't tell is if your mount has THREE or FOUR legs on it coming out. Most projectors have four mounting points, not three. So-so installers only use 3 of the included 4 legs on a mount because they are bad at their job. Kind of like not putting a brush plate on your wall to keep the installation clean where the cabling comes out.
Yes, the mount SHOULD work. If the mount is currently in ceiling studs, then it should support a hundred pounds or more quite easily. No projector should ever fall down if properly installed into a ceiling stud.

Be aware, that (once again) the AX200 was VERY versatile in how it could be installed. This included having almost the most lens shift of any projector on the market. If you have a 106" diagonal screen, and use the BenQ TK700, that projector has NO lens shift. So, the center of the lens MUST BE 5" above the top of the screen when mounted upside down on the ceiling mount. If you have a larger screen, it must be higher (by about an inch), if you have a smaller screen, it must be lower. Your mount doesn't offer any obvious height adjustment, so you will have to figure this out if you get the TK700.
If you need lens shift, you may want to consider the Epson 3200 or 3800 models instead.

https://www.projectorcentral.com/Epson-Home_Cinema_3200-projection-calculator-pro.htm

The Epson 3200 and 3800 have a fair amount of lens shift and zoom range which would likely be much more versatile for your installation.

3. The ONLY cable you will need is a proper HDMI 2.0 cable. Yes, your existing cable likely won't work if you actually want 4K to your screen. If cabling was installed during construction, conduit should have been installed to allow you to upgrade the cabling. Composite video is dead. Component video is dead. If cat-6 was used instead of the more likely cat-5e, then that's shocking, but actually useful.

You can't really put sources at the projector, because you have no way to get audio into your AV receiver. If your AV receiver isn't 4K compliant, then you will need to figure out how to get audio into your receiver separately from the video going to the projector.

A product like this will extend the video to the projector over category (5/6) cabling:
https://www.amazon.com/dp/B079PMQSZ8/ref=sspa_dk_detail_2

A product like this will give you the audio to your older 1080p capable AV receiver separately from the video to your 4K projector.
https://www.amazon.com/gofanco-Prophecy-Intelligent-60Hz-Splitter/dp/B079HJWF31/ref=sr_1_1_sspa

Or, just get a new receiver which supports 4K video properly to handle the audio.

It is possible, using the HDMI splitter I listed, to put a single source (AppleTV 4K) at the projector and send audio back across your existing HDMI cable to the AV receiver as an input source into the AV receiver.

What you want to do isn't really a straightforward upgrade, but if you do your homework, you may be able to get decent results. Jumping from 1080p (or less) to 4K rarely works as expected in projection setups. So, be aware that it may not go smooth and you may need additional parts to make it all work perfectly.

And one more question, this time about screens. When I redid my room some 18 years ago, I did it with the projector in mind—I had read in an AV forum that one could get by using Behr Silver Screen paint, and since my wall was freshly done and as flawless as a wall can be, I pained the entire room in that color and have had great success with the AX200U projecting directly onto the wall’s surface. However, I’m sure that will simply not cut it with one of the newer projectors (in fact, I’m sure the AX200U would have looked better with a proper screen and a dark bezel).

I saw elsewhere in these forums that you prefer Da-lite Progressive HD screens. Given that I have a white ceiling and light gray walls but am front projecting, what gain would I want the screen to be? The Epson 3800 is listed at 3000 lumens and the LS11000 is listed at 2500 lumens. A tool on De-lite’s site seemed to suggest a 0.6 gain. I was imagining that a 1.0 gain would be the way to go.

Also, the color of the screen—white or gray? Again, this is not a light controlled theater room, though it does have blackout shades and the carpet is a dark gray. But I can’t escape the white ceiling and the light gray walls. That is primarily why I’m wondering if the LS11000 is overkill or rather its advantages would not be noticeable in this environment. I am quite positive that the LS12000’s advantages over the LS11000 would be lost…

Thanks again for any advice you may have for me,
Oliver.