If you need access to the $200,000, you could use a HELOC to temporarily pull equity from the property when good opportunities arise. 

We’re planning to move to Europe to start our family, and we’re looking to generate $1,000–$2,000 USD/month in net income (after management fees and other expenses) with these RE investments.

If not you can close and simply pull out a 2nd or opt to take out a personal loan and consolidate on the ARV at the 6th month mark.  

If you were able to pull out $500,000 in existing equity out of a paid off rental and disperse those funds to acquire 2, 4-unit MFR properties and had the income to support any future CapEx and the interest payments on your HELOC ( including the extra income to paydown the loan ) would there be any " gaps " or problems I'm missing in this situation where it would be a bad enough idea to not go through with acquiring more properties through a HELOC on 1 of 5 properties you own?

You also need to account for taxes, insurance, HOA dues and management fees. 

@Dennis GallagherIt's my understanding that the "Income-Expense Ratio" primarily use operating expenses as the expense variable, which includes costs like utilities, property taxes, insurance, maintenance, repairs, property management fees, and trash removal, all of which are considered when calculating a property's operating expense ratio (OER).You calculate OER by dividing the total operating expenses by the gross operating income of a property.

Running detailed cash flow projections, factoring in their management fees and expected rent increases (you can use some of the great BiggerPockets Pro tools like the calculators to do this!)

Some landlords even add a small fee to cover these kinds of costs.I'd be happy to connect and discuss more! 

If there are only 2-3, then investigate and question GC.3) Check with your state's Attorney General's office for any complaints.4) Contact several City Building Depts in the area and ask for list of permits they've pulled.

Are you paying any utilities as the owner, aside from water and user fee?