Impact of custom ROM/root on Pixel 1 unlimited storage

Hello good folks of the pixel community,

I have an original pixel that I use as a photo backup device for my pixel 5 daily driver. I have a couple of questions if anyone can help:

1. Does rooting and installing a custom ROM cause issues for photo backup? I.e. will I still get unlimited original quality backup with a custom rom

2. My battery is shot and hardly gives me 10-15 mins of screen-on time. While I can get a new battery, is there a software bypass for the phone to directly use power from the wall and bypass the battery (something like ACC). Will ACC work after I root my phone?

Appreciate all the help and thanks in advance.