We run credit and criminal history on everyone.

exp is really trying to stop this kind of behavior! 

Since it doesn't sound like you're trying to catch any lease-violation behavior I'd give more notice than the minimum.  

Also be aware that being a European company they appear to be unaware of potential shady behavior in the US.

So I use a service where when the tenants fills out an application and pays it runs a full background check the main things I look at is credit and criminal history.

They can recognize patterns of behavior that may indicate dishonesty, such as frequent changes in contact information, suspicious rental history gaps, or an unusual number of co-signers.

There's nothing much you can do to screen the tenant as far as criminal background or eviction history.

I would create a short one page application to be filled out whenever new adults are moving in, and the company would provide you that and a nominal fee ($10, $20, how much do criminal checks cost in your area?)

All you need is a company to do the credit and criminal background search.

I'll give the apps a quick review and if nothing jumps off the page at me as a disqualifier, I then require the full security deposit from the applicant in order to move to the next step of actually running paid credit/criminal history checks.