I would leave the lights on for 10 - 12 hours.
You do not need a protein skimmer for a 10 gal. Weekly 1 gal water changes should be fine.
Many of us use a 100ml pouch of "Purigen" by SeaChem. It's a synthetic carbon that is suppose to work like a skimmer and is totally regenerable. You would put that right in the HOB filter.
Also for $10 you can get a
surface skimmer that attaches to the HOB filter and keeps the water surface free of any residual scum. Hagen makes one for the Aqua Clear line of
HOB filters that can easily be adapted to any number of tother filters. It even comes with 2 different size adapters.