How to choose a rootstock ? requirements, eligibility ?

What makes a tree a good root stock?
What are the factors for eligibility?

Can fruit trees be grafted onto native trees that are already established in this carribean forest?

I dont want to buy a rootstock. I have thousands of trees, how do I make a match?

It would be too nice (or maybe not) if you could graft anything to anything.
First you have to determine what species of trees you have.
Then research if they belong at least to the same genus (in some cases to higher taxonomy unit) of the scion you would like to use for grafting.
So for example if you have something in the genus Prunus  then there is a chance - but only a chance, that some other species from this genus could be grafted onto that tree.
It will require a lot of research on your side, but the project sounds very interesting.

I think it would depend on your goals. I don't prefer using conventional rootstock anymore but rather fruiting varieties or seedlings if I buy them. Or like you said if you have existing trees that would be fine also. As long as they are the same species they should work most of the time like Cristobal mentioned.

The qualities I look for are...
1) Disease resistance
2) Fast grower
3) Early Fruiting