I am trying to figure out how to fund my next investment purchase, but when I run the numbers for calculating a refi to pull out even just $40,000 of my equity, my cash flow seems to entirely disappear.

Have you tried the BiggerPockets calculators yet?  

View report*This link comes directly from our calculators, based on information input by the member who posted.

That pertains to buying more real estate especially an investment property that requires reserves and a DP that comes form a liquid reserve unless the lender allows for the use and the calculation of DTI with the new payment is within approved ratios.

I would also recommend using BP (Biggerpockets) calculator to compare the SFH and Duplex to see if the numbers work out for you.

. - The net profit calculated on all our deals is the net paid to

For the 9% calculation, you're using 2700 which would be $900 for all 3 rooms.

I think understanding how to properly estimate deals on properties (the BP Rental Calculator is a great starting point for this).

How do you calculate the cost of damage for nailing/drilling into joists?  

Use an online mortgage calculator to determine how much that payment will translate to a mortgage amount.