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If you were to app for the new card, I'd highly suggest doing that prior to closing a card since your risk of not being eligible for the signup bonus would increase by doing so. Doesn't mean you wouldn't be eligible if you did it in the other order, but I wouldn't take any chances.
They're not a hard pull; I'm not even sure they do a fresh SP as they're based on the pull that is currently showing as your Experian FICO 08 score in your online account. You can ask for up to 3x your current limit; at $60k they may ask for IV via a 4506-T. If you don't want to do that you can cancel out and ask again for a lower amount. (Then ignore the 4506-T form(s) you receive in the mail.)