Can the president of the USA run a Super PAC? Can they contribute to political campaigns other than their own?
No politician with a campaign committee can run a SuperPAC. SuperPACs are not allowed to coordinate with campaign committees. So no, the president can't.
Yes, anyone can contribute their own funds to other campaigns. Certainly politicians can donate to their own campaign (without limit). And they can transfer funds from their campaign to other campaigns. Anyway, a president can certainly make a regular donation from his own funds subject to the normal limits.
Note that a SuperPAC can't contribute funds to political campaigns. It can only spend money on advertising and what are called electioneering activities. Electioneering generally includes things like registration drives and get-out-the-vote activities.