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That's 3 very broad questions. I haven't seen any mechanism like that ready-made. The differential drive you mention is by far the closest one. I don't know if you have much more choice than to dive in and tinker with the parts.

If I was in your shoes and wanted to simulate that robot, I would probably try to find a working wheeled robot setup in Gazebo and change the links so it looks more like yours. At least you'll have a starting point. I can't recommend anything, but there are multiple ROSCon talks about racecars that seem to have shared their work. It would be helpful for future readers if you added links to what you find to this question (or answer it later when you feel more confident).

Note that MoveIt is for planning and does not simulate your passive wheels. You will need to use Gazebo for that (and don't expect much in terms of precision – contacts are difficult to simulate).