Again, I have 100% equity in the house.I was asked why I don't get a home equity loan and the reasons are 1) My credit probably shows me to be a "ghost" since I have not had any sort of credit in the last 7-8 years and 2) I have worked for myself for the last decade and my income is usually in cash so, it would be impossible to prove any consistent income.I never would have thought that getting a short term loan under my circumstances could be difficult to put together.

@Alicia Gates(SFRs). [24 CFR §110.10(b)](b) This part shall not require posting and maintaining a fair housing poster:(1) On vacant land, or(2) At any single-family dwelling, unless such dwelling(i) Is being offered for sale or rental in conjunction with the sale or rental of other dwellings in which circumstances a fair housing poster shall be posted and maintained as specified in paragraph (a)(2)(ii) of this section, or(ii) Is being offered for sale or rental through a real estate broker, agent, salesman, or person in the business of selling or renting dwellings in which circumstances a fair housing poster shall be posted and maintained as specified in paragraph (a)(1) of this section,

What should I know about purchasing the property under these circumstances?

you get a loan you're paying an interest rate for however long you have that loan. if at some point the circumstances of investment allow you to repay that  loan then great.

Do not under any circumstances enable her.

@Daniel Urban, a commercial property's value is based on how much money it IS generating, not how much it COULD be under ideal circumstances (there are exceptions, of course).If you're serious, I would determine it's value based on the information you have and generate an offer based on that.

However, given that we’re both military and a couple other circumstances, we have a bit more capital than a typical young-20s couple to invest so are also looking to buy a multi family (ideally quad or tri-plex) property that we can do some light rehab on and start cash flowing.

It will be hard to convince an underwriter that you are married living separately, although not impossible with the right circumstances. 

Ask more about the circumstances around the "eviction".

A third late will be a triggering of a eviction and the tenant will be out.A tenant that pays late under no circumstances should ever be considered a "otherwise good tenant".Landlords that have tenants that pay late have not properly trained them and are responsible for the problem.