You definitely need to make sure you take your time and plan things out properly.

First: RUN CONDUIT!!!
1.25" or larger conduit between the equipment location in the room and the projector location. HDMI didn't exist 20 years ago, it won't exist 20 years from now. First generation HDMI cabling didn't support 1080p, and 1080p cabling didn't support 4K... So, your cabling between the equipment and the projector will need to be replaced at some point in the future. You don't want to have to tear down sheet rock to replace a $100 cable. That turns a $100 installation into a $1,000+ installation.

Please read the FAQs in each section of the forums.
Also, please read: https://www.projectorcentral.com/What-Type-of-Cable-Should-I-Run-for-My-Projector.htm

At the end of the day, you will want at least a 18Gb/s HDMI cable between the equipment and the projector. You will NOT want to terminate the cable at the wall, but pass it out of the wall directly into the output of your AV receiver. 4K does not support passing through a wallplate connection (no matter what is advertised!!!).
So, plan on at least 25' or 30' for the HDMI cable to ensure you have enough slack to make it from the equipment, into the wall, up the wall, to the projector location, out of the ceiling, and down to the projector itself.

RUIPRO has some 8K fiber certified cables and they are considered a highly respected brand...
https://www.amazon.com/RUIPRO-Dynamic-Flexible-Projector-Theatre/dp/B081SQXPWB/ref=sr_1_14

So, HDMI plus 1.25" or larger conduit.
Note: Use Carlon flexible conduit with a pull string inside, like this: https://www.homedepot.com/p/Carlon-1-1-2-in-x-25-ft-Electrical-Nonmetallic-Tubing-Conduit-Coil-Blue-12010-25/207160098

That will take care of your projector wiring.
Please be aware that you should pick your screen size before you plan your projector location. 12.5' lens to screen (throw distance) is NOT VERY FAR! You will end up with about a 120" diagonal screen max. If you are sitting at 12' away, this may be fine, but if you decide you want to go a bit larger, like 133" diagonal, you won't be able to do it without moving the projector backwards a few feet (depending on projector model).

I guess my follow up questions are as follows:
1. How far away from the screen will you be seated?
2. MUST the projector go at 12.5 feet? Or was that just a random guess for placement?
3. How tall are the ceilings?
4. What colors will your room be?
5. What color will the carpet be?
6. Are you zoning up lighting and putting it all on dimmers as you should?

Yes, I would get a fixed frame screen over a retractable screen in a second. You get much higher quality for much less money.

If you aren't going with good floor standing speakers, I would figure out how good I want the sound to be and how much of an audio setup I want to get into. Audio can run from about $1,000 to $100,000+, and you often get what you pay for. I would probably get a new Yamaha receiver which supports (kind of) 8K, for the new gaming systems. On a budget, I might get Monoprice 8" 3-way speakers:
https://www.monoprice.com/product?p_id=6816

Depending on the wall size/screen size, I might put the speakers behind the screen and use an acoustically transparent (AT) screen.
I would almost definitely get a Silver Ticket screen. They have both woven acoustically transparent as well as standard white.
https://www.amazon.com/STR-169120-Silver-Ticket-120-Diagonal/dp/B00DGW15NU/ref=sr_1_3

Be sure of what you want before you go to far down the road of buying it. Separate out your projector budget from your screen budget and your audio budget and the rest. $1,500+ will go to audio, screen, cabling, and projector mount. If that comes out of your $3,500 top budget, then that only leaves $2,000 for a projector. If you have $3,500 for a projector plus money for audio/screen/etc. then I would lean towards the Epson 5050UB projector. It is one of the best you can get for a more dedicated space and at about $2,800 or so, it's a great performer with a ton of placement flexibility. You will want to place it about 15-18 feet away from any typically sized screen.

Learn to use the Projector Distance Caculator:
https://www.projectorcentral.com/Epson-Home_Cinema_5050UB-projection-calculator-pro.htm?id_=135

With a 135" diagonal, the lens to screen distance must be between 13'3" to 27'10". So, you would need a minimum distance of about 15' from the center of the mount to the wall for this to work well. Move the projector BACK!!!

Do not put equipment at the front of the room! Blinky lights don't belong there. Your speakers and screen belong at the front of the room exclusively.

With your room size, I would seriously consider doing a 7.1.4 Atmos setup. The room may support it nicely.

Draw everything out on paper, spend some time doing it all, wire appropriately.

Ask about specifics and take your time reading answers.

A lot going on here, so if I mess up or repeat myself, I apologize.

With your seating distance, I fully expect you to be very happy with a 100" screen. That will be quite immersive from your seating distance and should work well. In fact, I might hold off on buying a screen until after you have the projector in place and test it at that size from your seating distance to ensure you are happy with the 100" size or if you want to tweak it a bit larger or smaller.

Make sure when discussing throw distance to understand that the distance is measured from LENS TO SCREEN. So, if it is 12.5 feet from the beam to the wall, then your throw distance will be 18" or 24" less because of the depth of the projector. The lens itself will end up a couple of feet closer.

Yes, you can mount it further back, just behind the beam, for example, and have the lens just under the edge of the beam and have more variation in the size of the image. While I'm happy to point this out, you will have to figure out what works best for you.

HDMI Cabling:
Just as an FYI, CL2 is a fire approval rating which means you may run the cable inside your wall legally. It doesn't mean the cable is copper, or fiber optic, or made of a specific material. It typically just means that the cable has a proper jacket on it and may be used in walls. CL2/CL3 and other ratings are often used on cabling to state which type of walls they may be used in.
CL2, CL3, Plenum, CM, CMR, and CMP are all different types of fire ratings.
Hey look! A whole page about this:
https://monopricesupport.kayako.com/article/32-what-are-the-fire-safety-ratings-that-your-cables-are-available-in#:~:text=The%20main%20difference%20between%20the,the%20place%20of%20a%20CL3.

I'm not sure what copper cabling supports 48Gb/s at the length you need. I would not use them. I would buy a single cable from a reputable company with good reviews. RUIPRO has been one of the better rated cables over the years. I've used them a few times with very solid results on 4K (18Gb/s) content. No problems.

It is good to run networking cable between the projector and a network closet.

REMOTE CONTROL!
Hey! There are a ton of remote control options that you have. If you are putting your equipment off to the side, you should run a piece of raw Cat-6 cable between the equipment location and the screen location. You can put an IR extender right under the screen or by your center channel. That will take signals from your remote controls to the equipment location. Xantech is the best.
https://www.xantech.com/products/all/ir-kits/all

They aren't cheap, and you may want to try a cheaper version from one of the well reviewed kits on Amazon.

I use a RF remote in my home setup. It's very high end, so I can't speak too much about other systems that are out there, but when we need IR repeater kits, we always use Xantech because of their extremely high reliability.

If you want to get fancy, get a universal remote control. The Logitech Harmony remotes are still considered one of the best consumer programmable universal remotes on the market. I believe it uses both IR and RF to allow for devices to be hidden away more easily.

There will be a bit of work understanding how IR emitters work and should be attached to equipment and how to test them.

On budget, if you can swing the Epson 5050, then that's the way I would go. It's bright, has good contrast, and has extensive placement flexibility. It would be my first choice for overall quality.

The LG HU810 may be my second favorite choice in this price category. But, it is more suited to less ideal conditions. A very solid performer, but it doesn't have the contrast of the 5050. It may appear a bit sharper.

Moving the needle to a 100" diagonal, the Epson 5050 can have a throw distance between 9'10" and 20'8".
https://www.projectorcentral.com/Epson-Home_Cinema_5050UB-projection-calculator-pro.htm?id_=100

If we lock the throw distance at 10'6" you can throw an image size from 51" to 107" diagonal. So, it will work from your distance to the screen at the size you want, no problem.

As for lighting. Just walk the space a few times. Remember that NO LIGHTING should fall on the screen while watching a movie, but you may want light at the back of the room, or an adjoining space, which is good for that space, but doesn't fall on the screen. Just walk around and imagine where you want light, and where you don't, then ensure lights are installed which support that scenario.

You may find that having a solid surface flooring in the theater space is too bright. That is, reflective and noisy, creating audio that doesn't sound as good as you would like. It is very common for theaters to be carpeted to help deal with that. You can address that at any point you would like to.

I'm a big fan of separate rooms, even if combined, having a clear designation of separation. So, if the I-Beam is the separation point, then definitely have one side as dark as possible, for the theater, and the other side can be to your choosing as appropriate. A dark theater is a real bonus to getting the most from your theater.

It also creates a different space in your home that simply feels different. Very peaceful.