Air 3 has laminated/fused LCD and digitizer, tools needed to separate and rebond them are not cheap, it’s more economical to buy the screen assembly from a reputable vendor like MobileSentrix/iFixit/InjuredGadgets etc.
First things first: never buy iPad screen from Ebay or AliExpress! they are notorious for false advertising and low quality screens that do not match description. That iPad pro you have comes with a daughter board on the back panel of LCD, I’m not sure if it’s soldered on or not, if it is and you are capable of doing soldering work, then buy a standalone screen and transfer the original daughter board over, otherwise just buy one with daughter board pre-installed at a slightly higher price. I bought a lot of screens from MobileSentrix, base on my experience dealing with multiple vendors they are among the best if not the best iPad parts supplier in terms of quality/price/customer service etc. (your experience may vary, don’t hold it against me), search around and compare review and pricing across different companies, look for specific keywords such as tesa tape, daughter board, once you make the purchase and receive a replacement screen, take a few minutes to go through this fantastic guide written buy...
You shouldn’t be paying for ‘screen lifting’ issue as the onus is primarily on the repair shop, it was either a bent chassis which they neglected to straighten out, or they used a cheap adhesive that usually is the culprit for weak adhesion. I’d argue for a free warranty repair or at least refund of whatever price you paid for repeated work.
LCD and digitizer are separate on iPad mini 2 (meaning you can salvage either one given it is unbroken and functional), that long connector sits above short one is to connect LCD. Make sure LCD is still working, then all you need is the digitizer. Follow the digitizer replacement guide thoroughly on iFixit, and also go over the exhaustive guidelines written by @refectio in his answer to this question: After doing my repairs (battery and screen) I have problems
As far as I know, iPad 5 and 6 use different LCD screens (not interchangeable). I don’t know why iFixit lists them compatible, maybe I’m wrong. In any case, you should be eligible for an exchange/refund since the original LCD works but not the replacement, contact customer support and they’ll assist you with RMA.
Those are left and right signal antenna cables, MobileSentrix have them for $1 each: https://www.mobilesentrix.com/right-and-left-signal-antenna-cable-for-ipad-air-1 https://www.mobilesentrix.com/right-and-left-signal-antenna-cable-for-ipad-air