So let me get this straight..
Ledig = Free as in free schedule; available.
Fri = Free as in a free country; independent.
Gratis = Free as in cost free; free of charge.
Is that about right?
Yes, that’s correct. Fri is also used in other situations meaning ’not contained’ such as a free prisoner, a free bird etc, not only with countries.
Not reeeeally. Available = tillgänglig. There's overlap, but not entire synonymity I think.
Yes, if you want to book some service of any kind, e. g. at your garage. Then you can ask "Är ni lediga i morgon?", which means "available to take my car for service". Obviously, you don't want them to be "free" then.