Are there any problems with stuttering or the battery life on the iOS 12.4.1 update?

No troubles. Have you tried restarting your phone? See this link-->Restart your iPhone, iPad, or iPod touch - Apple Support

I had the same problems with my iPhone 6. This post helped me https://discussions.apple.com/thread/250604098?answerId=251156936022#251156936022 . Solution was no 3.

No troubles.

Yes, I did. On both my iPhone and my iPad Pro. Zero problems.

Yes. I did. Still have huge battery issues.

All iPhones and iPads in the house updated to 124.1, none with any issues whatsoever after the update. Use the link Lawrence provided.

Iphone 6S iphone 8 and ipad mini4 updated without any problems on any of the updates so far.

I always use iTunes in preference to OTA and keep them clear of unnecessary clutter.