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?

In reality, the termite damage we've experienced has always been contained to a portion (or two) of the house, was easily repaired with new framing, and wasn't a headache because we were already demolishing to the studs.In your shoes, I'd prefer to handle the remediation instead of the seller doing that work.

Is it fair to keep a portion of the prorated rent for the extra paperwork I'm doing? 

In many instances a large portion of the rent is almost guaranteed to hit your pocket each month - again, this may not cover the full amount but they are consistent payments coming in (paid by the government).

Sale the house, use a large portion of the money down along with borrowed money to invest in multi-family apartments?

If the PM has vetted the tenant correctly, they shouldnt have payment issues, especially if a portion is from housing subsidies. 

Also, the existing loan is already seasoned, and a good portion of the interest has been paid.