Wow! Glad you're doing good research (...and if you ask "What's good research?" I may have to find you and bop you on the noggin!

)!!
The best way for you to get good info on all that, is to do SEARCHES on all those questions, because there are LOTS of possible answers, and lots of good opinions, to each question.
I'll do a quick answer on most of them, but I urge you to look in to it more by using the search feature on the forum.
A general rule of thumb for a good dog/cat food, is to look at the list of ingredients, that are listed in order of greatest to least amount, and make sure that at least the first ingredient is MEAT. Other than that, some of us avoid meat by-products, but not everyone. Lots of people on this forum have fed their skinks the cheaper stuff for years and years with no ill effect. They just add good, high nutrient vegetables to the meat. Many add a multi vitamin and/or calcium powder every so often.
Yes, you can add calcium to the dog/cat food, mix it right in. Generally people say they "sprinkle" or "dust" some on and then mix it in.
How much/often is different for everybody. I don't think there is a "general" amount...? Some dust a bit on once a week, some in every meal, some not ever.
As far as a GOOD calcium, just look for stuff that is 100% calcium (or some with added D3), made for reptiles.
Again, the use of UV is as diverse as the use of calcium. Some do, some don't. Do some searches on the topic, google it.
You can take your skink out at any time as long as he doesn't get too cold for too long. You can hold him in your hands in the winter time just to let him get some sun for a few minutes, or when the grass or back patio warms up enough you can let him walk around for a while. In the dead of summer you'd have to offer him some shade. Just use common sense, and your own hand as a gauge for how long he can be out. If I put my skink down on the grass, I usually plop down too, and when my hands get too cold/hot, then I figure the skink might be too.
Hope that answered some of your questions.