Note that a skylight filter is very slightly pink, so will introduce a colour shift. A better option is a UV filter. They're not much used for filtering UV these days because modern lenses actually filter out more UV than the filters do, but they are completely neutral in terms of colour.
(Less well-informed shop sales staff will tell you that UV and skylight filters are the same thing - don't stand for this.)
I use a lens cap to protect the lens when I'm not shooting, and nothing when I am. Modern lenses have expensive, high-quality anti-reflective coatings; things like UV filters generally don't - so they tend to pick up reflections and glare much more easily than the front element on the lens does. The only time I'd choose to use one is in a specific situation where I expected rocks or gravel to be flying at the lens, and even then I might be more inclined to use a sheet of lexan or similar to protect the entire camera.
Into the lens? I guess it depends on the lens itself. Canon L lenses are environmentally sealed; as long as they are attached to the camera moisture isn't really an issue. I've used my 5D with L lenses in light rain without a problem. With older or less expensive lenses it could be an issue, but nothing you put on the front will change that - moisture could still get in elsewhere on the lens.
The main issue with humidity is to be careful when moving between air-conditioned indoor and humid outdoor environments; if the lens is cold it will cause moisture to condense on it when you take it outside. You should put the lens in a ziploc or similar before going outside, then give it time to warm up to ambient temperature before taking it out of the bag. Same with the camera body.