Can I finance an unlocked iPhone?

If you finance with either of Apple's plans, Apple Card Monthly Installments or iPhone Upgrade program (Payments), then the iPhone will be unlocked, but you need to enter an existing number with one of the supported carriers. It does not need to be your line, it's just for setup. Once you receive your iPhone you can bypass the phone number requirement and enter your own sim card or esim. T-Mobile is the easiest one to do this with.

This is what I saw when playing with an ordering page on the online Apple Store (in the U.S.).

• If you buy a phone outright (no financing), "your new iPhone will be unlocked".
• If you finance with Apple Card Monthly installments or with Apple iPhone Payments, "your new iPhone will be unlocked" – but you must initially select one of four carriers.
• If you finance with any of the carrier plans (AT&T, Boost Mobile, T-Mobile, Verizon", the words saying that the iPhone will be unlocked go away.

I would assume that if you finance with any of the carrier plans, your phone would be locked.

Sure, use your credit card to pay for the phone in full, and then make monthly payments to your credit card company. Or, take out a loan at your bank and have them set up automatic monthly payments from your checking account.