I would let the lease sleeping dog lie.....right now it appears they have no binding lease, so you have flexibility to put them on month or make them sign Your lease.

Also, if he did lie on his application, this may be grounds for termination of the lease as well.  

I'm active in Columbus as well.I have one or two referrals depending on where it lies on the C-D scale. :)Let's chat!

I have no need for this AC unit, but my question lies in "training" my tenants.

Screen provides just the right amt of real estate :-)I can’t lie,  I’m more lenient towards Apple too.

This would depend on whether or not the old landlord knew about the pet (or if tenants lie).

Assuming there are 6 people in the lease agreement already (maxed out), how do we know whether more people are secretly living there if they never notify the landlord and every time the landlord is scheduled to pay a visit, they can either tell that person to leave for the day or they can lie about it and say he’s a guest.

It may not require much money to get started in that but, depending how you were grown is to do usually requires lying to the seller it's not as easy as you thought. 

-Is this a breach of her contract because she lied?

Again, we're talking about someone (probably lying) about not getting an email copy of a signed document.