Can you boot to fastboot mode (Power + Vol Down)? If you can get there, you should be able to flash a normal recovery and get going from there.

There are issues, and yes they are usually fixed by transferring the NFC chip from the old display to the new one. iFixit also states (at the beginning of this guide: Reemplazo de Pantalla de Apple Watch Series 2): "Replacing the display can cause issues with Apple Pay. If possible, to reduce the likelihood of problems, delete all your ApplePay account info before the replacement, and then re-enter it with the new display installed." Could be worth a shot as well, I’ve never tried it. This was a Series 2 specific issue. I believe the NFC chip on the Series 3 and newer is a part of the main board, probably for this reason.

I’m confused. You’re trying to install Mojave on older machines? I would create a Mojave USB installer (https://support.apple.com/en-us/HT201372) and boot to that. I wouldn’t mess around with recovery mode at all.

Yes, that would do it.

The 2011 MacBook Pros were notorious for overheating issues, there’s no getting around that. Even a 2012 Unibody would be a significant upgrade for you, and you’d get Catalina along with the older design with the upgradable hard drives and all that fun stuff. But anyhow, back to this one real quick. Have you tried replacing the hard drive cable? The 2011 hard drive cables were total garbage. The 2012 cables were more robust, and they will fit in a 2011, so I’d just try one of those. MacBook Pro 13" Unibody (Mid 2012) Hard Drive Cable. I just had a 2011 here that was agonizingly slow with 16GB RAM and a Samsung 860 Evo SSD. Cable fixed it. If you’re set on a new laptop, I’d go with the Mid 2012 model like I was saying. Jumping from 2011 to 2012 doesn’t really sound like a big jump, but they were significantly better. You will need to purchase one with an SSD or install an SSD yourself, however. Any of the newer operating systems are basically unusable on spinning drives. Anything Late 2012 to Present are...

For this model I believe the power on pads (marked with a power icon) are located below the fan: You can just solder a basic button & two wires to those and you’re set.

No, they are completely different boards.

Occasionally this is just a software issue. Try powering it off via Settings (as opposed to holding the power button) and then rebooting: Settings-> General-> Shutdown-> Slide to power off. Alternatively, the home button cable in the screen backplate may be poor quality, and I would try replacing that or the screen if you’re able. Worst case you could have damaged the flex cable on the home button itself and that really isn’t repairable.

You will need the two tools listed at the beginning of this guide: HP Pavilion x360 m3-u103dx Fan Replacement Phillips #0 Screwdriver & Phillips #00 Screwdriver

You’ve likely damaged/shorted the logic board, which will require professional microsoldering repair.