If you replaced the screen, maybe it is defective or a cable is not seated correctly. You may also need to clear your fingerprint scans and re-enter them. I haven't tried replacing the screen on my 5S, so I am just guessing. I would try the settings before opening the phone up again. See if the fingerprint home button even works as a fingerprint reader. The home button only unlocks the screen if your fingerprint reads first.
get some graphite lock lube. Also, you might have some wiggle room if it's not the valet key. The valet key has thicker edges which block it from entering the trunk and glove box. There are many other solutions, depending on how much time you have, but start there if you just want the key out. If it's a copied key, the blades might be a bit rough so just keep wiggling the key with the steering wheel pulled and foot on the brake. Gently try to turn the key to the out position while pulling on the key a little bit. just don't force it, as it could twist the blades and make them lose alignment with the tumbler pins. if it's twisted, try un-twisting it.