Forget RapidVisa, so many sad stories on here who have used them.
Does sound like you need a Lawyer and the CR1 will be the best option.
You can marry, apply for an Immigrant Visa, you will need to be the Sponsor, of you can not meet the requirements than you can use a Joint Sponsor - Family or Friend is common. They need to be a USC or LPR.
Which is better depends on your circumstances. The only advantage (besides a rare situation with 18-20 year old children at the time of getting the visa) of the K-1 is it is a few months faster. K-1 is taking ~8-12 months. CR-1 is taking ~12-16 months.
The CR-1 is cheaper, lets you work almost immediately upon entry, lets you travel right away, etc. It also has a way to handle a refusal of the visa with a returned petition versus the K-1 (the I-129F for a K-1 dies if returned).
You can file while either, both, or neither of you are in the US for either option. Note there is a section on the I-129F asking for a couple more items if filing while the fiance is in the US.
In either case they would return home after any visit and complete the visa process abroad.
Yes. Sure he can leave. He can visit again (subject to CBP inspection) too.