Yes, to my moves and the smoke on water.

possibly yes to the fix. but the E sounds low to the A so possibly a higher action needed.

one other suggestion I have seen is to adjust the pick ups height.

I need to experiment and explore any differences using each of the pick ups and in combination?

If the string reads E open and at 12th fret then that would be intonated but action could have an effect at different frets.

More fun to be had

Dave