How was it decided who become a Regent if there’s multiple people vying for it? If there’s no will, would Jon Arryn just declare who he wanted as he was Robert’s Hand?

Well, that’s the crux of the matter: the Hand might have one opinion, but the Queen Mother might have a different one and if she’s got possession of the king that’s hugely influential, the king’s will might have some sway, the king’s surviving relatives might have another opinion, etc.