Includes the battery itself as well

“The current out-of-warranty battery service fee will apply until the end of February 2023. Effective March 1, 2023, the out-of-warranty battery service fee will be increased by $20 for all iPhone models prior to iPhone 14.”

Did the issue start after the battery repair? Repairs have a warranty, maybe they can replace your screen for free

How long have you had the phone for? The first couple days it can get hot because it’s downloading all your apps and stuff from iCloud in the background

Any chance it was already damaged when you bought it?

My general rule is to always stay updated as long as you have a backup (either iCloud or computer). Reason being that updates have a small chance of bricking your phone, requiring you to erase it and set it up again.

Updates help keep your phone as secure as possible.

Is the display a genuine apple part, or was it repaired with something else?

Check storage, try to get to at least 20% free space. When you say you’ve reset it, do you mean you erased it? If not, try Restoring it with a Mac or PC with iTunes. And set it up as new— don’t use a backup.

Also, you can call apple support and they can run diagnostics on your phone for free