The LL in the class C and above areas require credit checks (we also do a criminal check). 

Imagine the personal and criminal liability if there were every a fire in that unit.

There are a lot of great options out there; aside from providing the basics (ie. listing tool to get the property published to 10+ major sites, scheduling tool to schedule viewings, tenant screening with comprehensive applications which include nationwide criminal background checks + credit checks + eviction history, digital lease creation tool, rent collection tool, and maintenance tracking tool), some services also offer your tenants additional capabilities which makes it an easier sell on your end when getting them to transition over (ie. their ability to report on-time rent payments to the credit bureaus; this can create an added incentive for making sure they pay that rent on time).

I have started to play around with the software program Innago and see that they have tenant screening with options for criminal background, credit and eviction history.

My husband thinks using the persons name in the business name is tacky (no offence, he's not in rei)

LANDLORD is not liable to TENANT, guest, or occupant for personal injury, damage to or loss of personal property (furniture, jewelry, clothing, etc.) from fire, flood, water leaks, rain, hail, smoke, ice, explosions, interruptions of utilities, criminal conduct including theft, burglary, assault, vandalism or other crimes, acts of God, or other causes unless caused by LANDLORD’s negligence.

I also run a criminal background check and being here illegally is going to also violate that even if not convicted. 

If you lie, that could open you up to future liability; if you tell the truth, that could limit your buyer pool (a lot of people don't want to buy a house if work wasn't permitted);-  Worst case, down the road, if there is a defect in workmanship that results in someone getting injured or killed (electrical fire or gas leak, for example), you could potentially be held criminally liable and/or find yourself getting sued.

I rented her the apartment after doing my usual prior eviction checks, criminal checks, etc.

The whole section cited is a caution to lawyers to not try this kind of dual representation/simultaneous close unless a wide series of (improbable) things are done, and even then there is a section that says this whole process may involve other criminal issues.