Just installed new iFixit iPhone X battery. Now random shutdowns

There's a few things going on here.

First thing we have to do is determine whether or not the battery is the cause of the new shutdown problem, so you'll need to take out the new battery and put the old one back in. If the shutdown problem goes away then you've got a bad battery. If it persists then you MAY be looking at a deeper issue.

I definitely would have spent time addressing your original screen issue ("occasional screen shutoffs and darkenings") before swapping out the battery. By addressing the original display issue first you'd have only one variable (the display), maybe two (the old battery). But by swapping out the battery first you've now got FOUR variables to deal with (old battery, display, new battery, repair effort). Compound that with the new sudden-shutdown issue and there's a lot more troubleshooting that needs to happen before you can definitively nail down the issue(s) -- which at this point MAY not be a cost-effective repair.

The fact of your display's "bright spot" makes me think you applied too much pressure on that section of the display when removing or reinstalling it.

The fact of your hyper-sensitive keyboard could mean a damaged display. With the aforementioned "bright spot" I'm inclined to think this is the case, but there are things you can try first. You can go into Accessibility settings and adjust the touch sensitivity; that may be enough to solve your issue. Other things that could cause that issue are over-tightened screws (the ones on either side of the Lightning port), or misaligned and/or damaged adhesive.

Performing a factory reset on your iPhone to see if the issue is software-related is also possible, though I will say I've seen iPhones that were poorly repaired but generally functional (like yours) that were subsequently bricked after performing a factory reset. Caveat emptor.

As far as a refund from iFixit is concerned, you can ask them but I suspect not -- assuming the new battery is good. If the battery is bad, sure, they'll replace it, but if it's good probably not. You've already used the part and it's not their fault that your phone is having issues after you attempted your repair.