So basically, I've flashed a rom onto my phone, specifically lineage 15.1, for the reasoning of I like Android 8.1. The problem, however, is that my gapps seem to not work. Google Play services stop respond legitimately every 5 seconds, and I've seen all the stuff about force stopping it and cleaning the cache, but those options are greyed out for me. My phone is a Galaxy Note 3, if that makes a difference. Also, it wants me to do the Pixel setup, but it checks for updates continuously forever. What to do?
Did you do a "clean flash" (= wipe /data partition) when flashing your ROM ? Or did you do a "dirty flash" (do NOT wipe /data partition) ?
If you did a dirty flash, my advice is to do a clean flash instead, then reinstall LineageOS + OpenGApps (pico version as advised by @Rahul Gopi). Be aware that it will reset your phone (non-system apps, settings, etc. except your USB storage/card).
As breversa says, you should do a clean flash. However if you don't want to loose all your non-system apps and settings you could first try to wipe /system and then install LOS 15.1 again, flash the Gapps after and clear (Dalvik) Cache, no need to wipe /data. If that fixes the crashes, great. If it does not, you should indeed try a clean flash.