Would I need to get appraisals and renegotiate the entire loan I assume to pull out the one I want to sell before I'd be able to do so?

If you do a cash-out refinance and pull that equity out, then you'll be paying interest on it all the time because you've permanently pulled it out (until the loan is fully paid off).If you do a HELOC, you'll only be paying interest on it while you're using it.

I had my broker pull up the recent average days to rent in my area/ my apartment size/etc.

You can take it one step further and ask a realtor or someone with MLS access to pull which zip codes the most cash buyers are looking if you want to find hot properties to flip and sell fast.

I know he doesnt think of me at all [I am a single parent of 2 doing my best & pulling from every direction I can to move up in the world]. 

This doesn't even consider the overhang on the trailer from the rear axle to the back of the RV, so there is a tail swing factor when backing into or pulling out of these units. 

Will the Rehab that you have effectively done make up for a market correction thus allowing you to pull equity from the property to properly Repeat the process or do you just have to wait for the market to bounce back before you can refinance?

I made him pull over when I saw the For Sale sign.

If you have any cash flow after paying the bills and setting aside funds for future expenses, you can either pull it out or (preferably) hold it for a future purchase in the checking account or in the savings account.I prefer holding funds in the savings account, even money for future purchases.

Refi is preferable if the money being accessed is for longterm purposes (ie buy/hold) and you plan to purchase another investment soon whereas a Heloc is preferable for short-term use (flip, rehab, etc.) and/or you want the ability to pull the trigger in the perhaps distant future if/when a deal presents itself.