@Portia Bates To answer all your questions:
It depends on the timeframe. If you have a lender already lined up who you want to work with then you can probably have the refi done in little as two months. It also depends on whether or not you have to wait the seasoning period which could be 6-12 months.
But the most important thing you must account for is the order of the BRRRR process buy, rehab, RENT, Refinance, repeat. You want the property generating income before you take on debt. It would not be fun to have a mortgage payment coming in and no income, not to mention the holding cost (gas, electric, taxes, water, etc.)
The quickest way to move the process along is to start working with your lender ASAP. Providing all the necessary documents beforehand helps a lot! The refinance process can take as long as you and your lender make it take. I will be refinancing a property in a week but I started uploading documents months ago to get it out of the way. I’m waiting the 6 month seasoning period. There is also the delayed financing program that @Alexander Felice preaches about. If you put your rehab cost on the closing statement (make sure you listen to podcast 301 and read his article on exactly how to do it) you could get your money back even sooner with no seasoning period. On average, the refi period usually take 30-45 days (as told to me by my lender).
As to the hard money question, that’s up to you.