I have never used the split or merge mechanic with a resource, so I have a few questions.

First, if I play Batman with the Batcave assigned to him, and then merge with Robin to form the Dynamic Duo, does the duo keep the Batcave.

Second, if I play Dynamic Duo with the Batcave assigned at the start of the game, when they split, who gets the Batcave.

Finally, am I correct in assume that in situation one, if the duo later splits, the resource stays with the Batman.

Thanks.

1) yes
2&3) you choose

If a figure assigned a resource merges into a duo, the duo also gets the resource.

If a duo assigned a resource splits, you choose which figure gets to use the resource. It doesn't have to be the same figure that had the resource before if the figures previously merged.