I’ve filled out some refinance applications and have found out that with the increase in value of the property over the last few years among other factors, I am able to get about $40,000 - $50,000.

Since you only intend to do one deal at a time (now) understand the job job of your mail piece: it's ONLY job should be to get the prospect to take the next action step, such as call you, write you, go to a website or landing page, etc.There are so many factors to consider. 500 mail pieces is statistically too small to really derive any actionable feedback.

Any major factors here to be concerned with as I model this?

What percentages do you use or what other factors drive your rent increases?  

Since REI (buy and hold) can be a long term investment for many (5,10,15+ years) I believe it's imperative that you factor in time value of money into any analysis.

Again living at the location, so please factor in that Nashville has a strong sellers market and I do not want to live in a bad area or really far from town.

Agreed, buyers are most often looking for the classic 3/2 although here in Florida with the snowbird factor and empty nesters/retirees there is probably a larger market to sell 2 bedroom homes than other places

If you need to factor in property management costs (around 12%) for a rental, do any of you look to make up the difference somewhere else in the deal for a property?

One of my criteria is to get at least $150 cash flow a month on this rental, so I'm worried/curious to see how this increased mortgage payment may/has affected cash flow (leaving out other factors I need to account on the income and expense side). 

What are the important factors I need to consider when buying a co-op?