We live in the 2/1 portion of the home and rent out the 1/1 attached efficiency for $850.00 a month.

Frankly, I think the other side has an argument for asking for a portion of EM.I'd ask your lender to provide you with what regulation prevents them from having control over an appraisal timeline.

With the "BRRRR" I'm wanting to know down the line how would the refinance portion work.

For that portion of your income, two years of receipt will be needed. 

The way they look at income standards is based only on the tenant-paid portion.

So, I tracked down the owners and asked about buying a portion of it to augment my yard.

As always, caveat emptor with foreclosure/trustee sales and make sure you’ve consulted legal advice re:title and have physically inspected what portion of the property you are legally able to inspect.

We have a Section 8 tenant who's portion of the rent has bounced.

Most leases charge a total amount for the term rented (i.e, $12,000 for a one year lease) and then divide the payments up into equal portions for the benefit of the tenant making regular payments. 

Being an installment sale I know I'll pay tax only on the portion I receive each year but is this a very bad idea?