If you turn off Battery Optimization, yes. Apps can even draw over and control other apps. Assuming the LAM is just an app controlling AOSP in a VM server-side, this really should have just been an app with the LAM sending touch events back to the device.
Not sure about nowadays, but a few years ago yes. You could run a foreground(?) service which kept an icon in the righthand corner (can’t remember too well). Anyway, I had an app which kept a http server always running.
