I almost always have fritz running in the background (obviously never use it while playing) and I've played over 2k games so I would definitely say they don't take that into account while detecting cheating. Maybe if you switch windows constantly that might raise a flag. I know some sites do that. Not sure though
Go ahead, stretch the rules in your way. But if you therefore get banned spare us the details if you don't mind.
Cheating is receiving assistance from an outside source.
If an engine is running in the background and you never consult it, obviously that is not cheating. The question remains, however, if lichess somehow tracks background processes in your computer and that information is considered by the anti-cheating algorithm. I think that it is highly unlikely that this would be the case, but only a developer could say for sure.
If you want to be 100% safe better turn the engine off while playing.
