I'm wondering if Apple stores even still sell the MagSafe chargers any more; there might be some difficulty in getting a new one now? I think it would be a MagSafe 2 cable and connector. OK, just took a fast look and they are no longer available at the Apple store online and so would not be possible to buy as new in a retail Apple store. That said, though, the stores would have MagSafe 2 cables available behind the scenes in order to check someone's machine and for the Genius Bar personnel to do any simple repairs. The same would apply in a shop authorized to repair Apple products.
Can't recall now whether the battery in a 2012 MBP is user-replaceable, but it does seem likely that it is actually the battery which is the problem.
Yes, the older iPad uses the lightning connector/cable like the iPhone; the most current iPad Pro uses an 18W USB-C connector. I have at times actually used the 29W USB-C cable/charger from my 12" MacBook, too, and it works fine with that. I have not done the reverse and used the 18W cable/charger with the 12" MacBook, though. One needs to consider two things: whether a connector will fit into the charging port on a machine and also what the wattage of the charging plug actually is, as there are different requirements for different MacBook Pros.