We bought it for $225k with a 3.125% mortgage and 1) are interested in keeping it as our first rental. 2) We are thinking about using some of the equity (maybe $50k or less as a HELOC to combine with existing cash for our next home down payment that we can convert to a HELOAN through our credit union).

To do well, build a strong money base, keep a good credit score, and look at other loan types besides FHA.

The bank will want you to be working for a certain amount of time and that have certain credit score plus have your financial ratios match what they want. 

The good thing is, she is now researching credit unions and banks for a car loan.

There should be other options, however with an LLC and fourplex - yes, DSCR is by far most likely to be your best option unless you do the heavy lifting with local banks and credit unions (and these are increasingly not great options for this type of stuff as they move painfully slow and are under lots of regulations and restrictions)2) I haven't yet reached out local banks, but seeing the quotes from brokers makes me wondering if Banks may be better? 

She has a higher FICO score but full of eviction records which I was aware after she moved in and was late for the first time by using another public data source.

One of the things you can start doing is reviewing your credit score and work on improving it if needed.

What documents do they require, what credit scores do they allow, how do they verify previous rental history, etc.?

:Class A Properties:Cashflow vs Appreciation: Typically, 3-5 years for positive cashflow, but you get highest relative rent & value appreciation.Vacancy Est: Historically 10%, 5% the more recent norm.Tenant Pool: Majority will have FICO scores of 680+ (roughly 5% probability of default), zero evictions in last 7 years.Class B Properties:Cashflow vs Appreciation: Typically, decent amount of relative rent & value appreciation.Vacancy Est: Historically 10%, 5% should be applied only if proper research done to support.Tenant Pool: Majority will have FICO scores of 620-680 (around 10% probability of default), some blemishes, but should have no evictions in last 5 yearsClass C Properties:Cashflow vs Appreciation: Typically, high cashflow and at the lower end of relative rent & value appreciation.

I'm giving a reverse example, as in I will show how my credit score dropped 59 points in one month because I forgot to do my "trick".The trick is to pay off your entire balance the day of your "statement closing date".