If you were planning on it, then I bet you should be able to find a lender that can cross collateralize.

Of course, there are exceptions, but I think that's a pretty safe bet as a general rule. 

If your LLC refinances into a commercial loan, a wire will be sent to the old lender paying the old mortgage off in full.

Just make sure to ask for separate releases so you can sell or refinance a property out of that portfolio.

My first question, would I be able to do a cash-out refinance on my Colorado rental, to use as a down payment for another investment property, if I have maxed out my VA Loan entitlements? 

You have some equity, and you are looking to refinance.  

Universities are going to make money, your best bet is to play the game with them and sometimes you can't b/c they simply control the market too much.

I plan on using the BRRRR strategy for this property but I do not think that the ARV will be high enough for me to pull out all of 20% down payment after I refinance (I'm also content with leaving a few thousand in).

Now if you’re looking at $100k they may not have anything for you but if you’re (or anyone) is looking to place $1m+ I can bet that most commercial developers will have a project for you to come in one.

It's likely that in the divorce decree part of the agreement was that she had to keep the mortgage current, refinance, or something along those lines.