If you allow the jasmine up to the tree, it will climb the tree requiring at least monthly trimming. Jasmine spreads slightly slower than bermuda but it is equally persistent. I'm not saying not to do it - just letting you know what happens up against a tree. I've done it both ways and prefer to keep a border away from the trees.

There is a saying that applies to many ground covers: the first year it sleeps, the second year it creeps, the third year it leaps. My experience with Asiatic jasmine is that it never really leaped. On the other hand, my next door neighbor installed surface drip irrigation before putting in jasmine. Theirs leaped the first year.

I would not necessarily pull the mulch. I think jasmine will grow up through it with no problem.