I'm looking to invest in solid countries (politically wise) that allow foreign investments, which makes me think I should look at the different markets in the USA.

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It might preclude your future personal involvement.There is an alternate structure known as a Rollover as Business Startup that allows you to capitalize a new or existing business with retirement funds, and be directly involved in the business.  

My one question is, in your experience, if you have had to change your financing from conventional to cash, does HUD still allow you to close w/in 45 days?  

Few, if any, conventional lenders will allow you to finance the down payment with another loan.

Or should I aim for whichever position that will provide a quicker path to a higher salary, therefore allowing me to actually invest in projects?

Yes that is a great strategy.As for the HELOC with the exception of huntington they really only do it on the primary so if you move that messes it up.However, Huntington will allow you to do a HELOC on one rental unit so I would go with them for that.

There are often issues on units that allow 3 singles but only two parking spaces.

A preliminary Team includes Attornies, real estate Agents, CPA's (or accountants), Inspectors, General Contractors, and Handymen.Always remember to do a math analysis and try to look at prospective properties before you make an offer or at least have a clause in the sales contract that allows you the time (about 15 days) to make a detailed inspection and possibly back out of the deal without a penalty that will hurt you very much.  

The ROBS 401k will allow you to use all of your retirement funds to invest in your own real estate operating company and draw a fair salary.