I'm fine with whatever general setting, though I do lean towards the fantasy side. Not neccessary the medieval kind, just the kind where the story isn't hyper-married to reality. We could have each universe run on their own world laws, but I'm more inclined towards there being general laws for the sake of simplicity, and so the universes can actually interact with each other.

As far name...depends what the plot/whole thing will be a about. 'Multiverse Mash Up' is a simple one if you need a name for the planning phase.

2 and 4 together sound fine and dandy. We could do each universe finding the others on their own, our characters being the brave explorers. Alternativly we could start where they've already integrated with one another, or at least recognise and have come to terms with the others existing. In that case, perhaps the characters could be a part of some multiverse...coalition? Combined force? That deals with something or another. Could be something specific like multiverse police, or something more broad like being apart of the organisation that organises/handles affairs between multiverses.