Issues with screen replacement. Need help addressing repairman.

Hello! I recently broke the screen on my LG G3 (Verizon) by dropping a mug on it (for reference, I got my phone in November). Since the insurance offered to us by Verizon required a deductible larger than the cost to purchase a new phone, we decided to find someone to repair the screen. We settled on a repair kiosk at the mall (Accessories, Repairs, etc) because a replacement is a replacement, right? Well, not exactly. The phone was repaired yesterday (entire LCD changed) and I've begun to notice some issues with it. I have a list. I contacted the repairman and he claimed that repaired phones never function as well as they did when new, but he'd be willing to look at it to see what he could do (for free). And so, I would like help with diagnosing the issues with my phone so that I don't show up empty handed tomorrow, and risk being turned away.

These are my problems:

>Physically, the back panel on the phone doesn't seem to fully connect (see picture)

>There seems to be a lag when swiping. I can't Swype things as fast as I used to, and the response is slower. Also, when swiping or when pressing with two fingers, other parts of the screen respond or freeze up. Ex: "Hey, what's fowwwwwwwwwwwwwwwwwwwww" when texting "for dinner." The extra letters appear almost instantaneously. Also, some things will remain pressed even after I lift my finger (like when opening apps). This is majorly annoying (and hinders my ability to play rhythm based tapping games).

>There appears to be "wrinkle" in the screen, as if the screen were made of cloth. It reminds me of the days when we weren't allowed to touch laptop screens because it would end up looking like that.

>The vibration seems misaligned. The vibration for texts isn't as solid (also it's louder), and button presses produce a very sad, shuffling feel. When in a call, the front speakers make the caller sound "robotic"

>The display lags when rotating

>The screen heats up after extended use (used to be only the back buttons)

It may seem like a lot, but really it's a collection of little things that, while not hindering the use of my phone for things like texting and calling, make my phone less pleasant to use. I have some experience with repairing my own electronics, but because I use my phone so much, I would rather it turn out perfectly than risk doing it myself. After doing some research, I noticed that most of the problems I have can be seen in iPhones after screen replacements.

Please help me diagnose my phone's problems, and come up with a solution for how to proceed. What should I tell my repairman? Should I find another repairman? Or perhaps try to fix it myself (I have no tools for this). Thank you very much and I hope the responses are swift.

Some helpful photos:

Also, you may notice that the color is different than the original. The new LCD is not the Verizon version. Would this affect anything? I've also learned that my repairman has already recycled my old LCD.

Thank you!