I need some suggestions as to what pixel height/width and dpi settings to use for my backgrounds. I plan for them to be highly-detailed using Coral Paint. Does anyone know what settings the pros use?

Well, depending on your project settings and zooming needs…
Rule of “Thumb”…? = at least 2 - 3 times your project settings…

Just working on a project with a bitmap background…
Project settings: 1920 x 1080
Imported image size: 5616 x 3744
(using just reasonable zoom in and outs)