I've removed all the strings a couple of times on some guitars to give the fretboard a good doing. Should be fine as long as:
a) you evenly tension down the strings across the neck a bit at a time.
b) your trem isn't recessed (sunk into the front of the guitar). this doesn't make it impossible actually, just harder and fiddlier.
if it's recessed it harder to shim it. if it's floating like a PRS trem, which i sometimes see floyds set up like, i using playing cards to keep the trem in position, just grab a deck, set them against the back of the trem and push, whatever the trem needs to shim it will get pushed in, whatever it doesn't stays behind.
If it's set up for down bend only, i push it forward and slide a couple cards in there anyway. can protect the finish from denting when string tension is removed.
You can also slack off the truss rod a little, but i don't, like i said, IMO as long as you slack the strings off evenly across the neck a bit a time and only do it once in a while, there shouldn't be any problems. just keep the neck wiped off properly and it should be long long amounts of time between when you need to do this.
As long as you string up again with your usual brand and guage of strings, it shouldn't affect the setup
For a good cleaning i get a very soft/old toothbrush, and lighter fluid (is fine for open grain or sealed necks).
I squirt some into the toothbrush (not the fretboard) and get scrubbing (lightly) with it. then i put a little bit more in a lint free cloth and polish off the board.
THEN i treat it with lemon oil or a fretboard conditioner. lemon oil, or what we call lemon oil, is fine, but IMO don't think there's any point in putting a treating agent onto something thats dirty. It'll just spread the dirt around and work it further in. it's like putting moisturiser on but not washing your face first (it's the 21st century, men use moisturiser ok?)
this is totally IMO though
