As an example, I purchased a 40k 3/1 SFH and looked on the county appraisal site and saw that the last time it sold was 20 years previous for the same $40k. 

Do you need funding for the rehab or just cash out and put the house on a mortgage after an appraisal?  

Using a cap rate calculation along with an estimate of operating expenses (30% operating expense margin as a soft rule of thumb), we estimate the building will be appraised at around $1.45M to $1.5M.

My question is, upon getting the property re-appraised after strategic renovations to increase the value of the home (Kitchen, Bathrooms, Flooring, etc) would I need to refinance to get the PMI removed or would it simply phase out as I should now have more than 20% equity in the property if everything goes according to plan.

The reports are 40-90 pages long and go through the entire house in great detail.Get an appraisal.

Is the value just apparent to the appraiser?

The financing available to me, less than appraised value What was the outcome?

You may want to ask your insurance company and lender about it as well as some insurance companies are now inspecting and may require a new roof if it's that old and not done correctly with the 3 layers, or appraiser may call it. 

We are looking for recommended appraisers, inspectors, and property managers.Do you know anyone?

Do you have the EM, inspection and appraisal costs, closing costs and down payment for both?