This particular rental, though, has been vacant for a little over two months...the longest we've ever had a vacany, but it's a unit above our house, and we've already turned a few people away because of criminal history or whatever reason.  

Such as, if the tenant; doesn't pay rent, fails to vacate at the end of the lease term, damages property, violates a condition of a written lease, is involved in criminal activity, etc.. 

For your criminal part, establish what your requirement will be:-No criminal history-No felonies-Misdemeanors (case by case)?

Take this as a lesson to ask for appraisal earlier in the year or fight the up hill battle against county government criminals?

Unless otherwise provided by law, proof of violation shall not require criminal conviction, but shall be by preponderance of the evidence and upon Landlord’s reasonable suspicion and at Landlord’s sole discretion.

Check their criminal history make sure everything checks out.

Was a criminal background check done?

You can find every criminal offense all the way down to minor traffic violations.

I’ve seen that service advertised for $400 and up, which is nuts, considering, in the case of a lease-option, it doesn’t include the criminal background check, eviction check, previous landlord checks, visit to current rental, and probably not the employment verification a smart landlord does.

There's no special process for criminals, the missing or the deceased.