Typically in my (limited compared to yours) experience most issues come up shortly after moving in.

The IRS considers debt reduction (when not replaced with additional equity in a new prop) to be the same as cash in terms of its benefit to you, so you would be taxed on the $125k you didn't roll over.If you don't want to carry a mortgage on a new property, you could sell, use $125k to pay off your current loan, and then put in an additional $125k of your own cash to make up the difference, resulting in $325k of equity in the new property/properties, and meeting both the equity and total value rules.So the short answer is YES, it is allowed for you to go from a $325k property to a $200k property while paying off your $125k loan, BUT you will pay taxes on the amount you don't roll into the new property ($125k).

In particular, it seems like working with a private lender / hard money lender with a short term loan for the course of the flip might do the trick. 

(this is important if you do need a short term loan to purchase then refinance out - and the answer should be 1 day...very important that it is 1 day on title is all that is needed to refinance) What is my minimum down payment required?

I do believe in 30 year loans as opposed to 15 year loans on short term rentals because I'd prefer to cashflow first then refi or sell within a year or two.For everyone else: What was your LTV and interest rates for your first investment property and how many banks/lenders did you talk to before finding one that best suits your situation?

One thing to keep in mind is property management for short term rentals. 

As a landlord, I'm more put off by your short term lease than the dogs (we allow them, excluding bully breeds).

I am hoping to be investing and living in Portugal shortly. 

banks have to hedge to ensure they dont lose money when rates go up. hence they charge for rare extensionon new constructions its preferred not to lock rate until its ready for possession. if you dont need to pay for rate lock extension if you walk away, suggest you go with new bank. 2 weeks is too short for a large bank to get ur loan ready to close.

I think she would take a partial cash offer, if we could pay off the rest in a short amount of time.