If you apply for a mortgage I believe that you have a 30 day period where you can apply to multiple mortgages where it would only count as one inquiry. 

Would it make more sense to purchase one property like a cash deal, purchase multiple properties with the private money as the down payments using the BRRRR method, or purchase a property using all of the private money as a down payment(then probably BRRRR with it)? 

Sep 2013 - Single Family home - Short sale/MLS - Short sales with other buyers fell through multiple times, we came in with a cash offer and got the deal. 

After I graduated, I got my real estate license and then got a soul sucking J-O-B as a leasing agent and quickly worked my way up to managing 100's of units and multiple buildings as a property manager.

@Alejandro Calixto VA loans won't fly on properties that have multiple compliance issues.

At this moment, I'm saving some capital for my next purchase, and I have also been playing with the idea of selling my current McAllen rental in order invest the equity in a 4plex on multiple properties.What's your opinion of the current housing market in the McAllen/Mission/Edinburg area?

You want multiple offers and the chance to pick the best or to counter for more or to do a round of highest and best, whatever is the norm in your area, but in general the more exposure the higher sales price.

It is complex for at least two reasons:Regulations can vary along multiple dimensions such as asset type, investor type, buyer jurisdiction, seller jurisdiction, and issuer jurisdiction.

The last option I can think (outside of sending mailers to distressed homes) is buying a single family home in one of the nearby tourist destinations. 

If you're self managing (I don't recommend) then it should be close to youSingle Building or Multiple buildings ok?