Your assumptions are correct, large valve is the black tank, small valve is the grey. What dumps into each tank can vary with different campers, and good to know. Since your up and open, close black tank valve, grey open, run water at sink(s), shower.
The other 2 lines you mentioned I don't see. If they are small, 1/2-1"
likely hot/cold water line low point drains.
AS to emptying tanks, again your assumption correct as far as connecting sewer hose. Normally though if you have sewer hook ups, the black is left closed, dumped when needed, otherwise water runs out/solids can stay behind. The grey though, dumping directly on ground as mentioned most places is not allowed, certainly not at any campground. Don't. It can be left open when connected. Ive never used any supports, we rarely leave or have hose connected so cant recommend anything.
Just using holding tanks, most dump the black then dump grey.
Length of hose 15' 20', I carry 2. Many campground site-sewer connections are simple pipes stubbed up-you'll need an elbow/funnel to use.
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I use Valterra 2 piece with ell attached-only used the 'funnel' couple of times. Even at dump stations, Stick ell in sewer drain-rock to keep from coming out.
Never pulled RV toilet, can't help with floor...Don't fall thru?
unrelated-has anyone mentioned the max photo width?
-640 recommended 800 max