Generally, a custom designed character you put effort into making + hold an auction for the rights of using the character / the new owner able to do what they see fit.
There's also cookie-cutter palette swaps of the same base.

The first mentioned is the way to go, if you wanna sell for a higher price.

Ok, say I want to adopt out these two characters from some older artwork I did: http://www.furaffinity.net/view/56189/ Can I do that?

I'd say so, yeah. Nice work, and whoa, 11 years back.

YOU LIVE!! *glomp*

Adoptables, they can be many things.
You could produce finished pieces, individual or cookie-cutter types with varying colors and patterns or features. Give some wings, some horns, some hooves, two tails etc. Just for some variety.
Another popular kind of adoptable is the egg adoptable. You paint up an easter egg, your client buys the pattern and then you apply it to either a requested species by the buyer, or my preference, surprise them with whatever critter you want to create but use the color scheme of the egg.
The buyer gets the rights you see fit to put in place, typically they get total control of the created char to do with as they want. Except when it comes to re-selling them. Most adoptable artists put the rule in place that an adoptable cannot be resold for more than was originally paid.
Many like to be credited when the finished adoptable that they drew is posted anywhere.

Can I turn older artwork characters into adoptables, like these guys? http://www.furaffinity.net/view/56189/

Oh yeah, unused chars make up a considerable bulk of adoptables.
Chars that you dont want to use anymore, or have never really used, someone'll take a liking to them I'm sure.
When they're established, and there is other art of them, then things can get a little awkward, but if they're mostly unused one-off pieces, yeah go for it.

oh so anyone can do adoptables if they use a certain art style of stretchy cartoonist beyond reality characters?

Interesting. I have a whole bunch of characters that I created just for a particular piece of artwork. I could turn them around and sell them..... Hmm.