My primary concern is that with me being out of state I do not know first hand what is going on with my rentals and my PM has lots of "small invoices " that are adding up!

If and when my tenants leave, I would go to the bottom of the waiting list to rent it out again and thereby be sitting on a cash drain with the mortgage and HOA dues, and I would not even consider moving out of my primary residence to live there due to the space my family needs.

I have been advised by my title agent that if I were to go through with the closing, I would be signing some untruthful documents regarding it being my primary residence since there is currently a tenant.

Its primary use is not tenant screening, but you can use it to manage any workflow.  

If you are buying as a primary, you have 60 days to move in. 

I want to take a HELOC out of my primary home to use as a DP for another investment property.

Really most of our purchases have been investment (non primary resident) and a couple of times we used conventional loans that had a better rate and came out cheaper than VA loan after their VA funding fees...

They are common for years in other parts of world (I see many in Europe where the heating is often primary radiators (no ducting) with Splits providing supplemental heat and AC). 

Do you have the ability to use it as your primary residence and attempt a Harp loan?

Yeah no short term loans for STNL unless you are putting 50% or more down or a lease at a high cap rate with a few years left before option renewal.Otherwise you get hit with a refi bomb that could make a negative return with a fixed rate lease or one that has minimal increases in the primary term less than annual inflation.People get these short term loans to massage the debt to work at lower cap rates for purchase but they are ticking time bombs.