I'm a bit intimidated at the thought of bad tenants, but I think proper screening will put me at ease eventually.

There is so much info here it is hard not to stay glued to the screen :)I have lived in the Bay Area my whole life and in San Jose for 45 years.

It generally just means that you cannot outright refuse to rent to Section 8 tenants simply because they participate in the program and that you should screen them like any other applicant. 

I vet buyers with owner financing very similarly to tenants so this guide will be helpful to you -http://www.biggerpockets.com/renewsblog/2013/01/27/tenant-screening/ 

Best I heard was to establish your screening criteria and stick to it.

There are several tenant screenings services online that can directly charge the applicant a fee, so you don't have to.

I believe some states/cities have protection for hoarders so you want to know about this too.If you do get the property make sure you screen any new residents, and that you have a clear policy on your rental qualifications. 

Can a diligent screening process make these properties work? 

Unless you can hold your contract up to the screen for us to read, I'm not sure what you signed.  

I can take out the drain plug and put in a mesh screen stopper instead, but it doesn't look good.