Even with my two month vacancy last year, I had two "near misses" from tenants that ended up not showing up to sign the contract and pay the deposit after verbally agreeing to rent after an initial showing.  

One time an angry tenant (he thought he was getting his entire deposit back that day) ran after my car spitting and swearing while my youngest child was with me.  

Make sure you get all the deposits too. 4.)

Once my wife was processing applications, said one guy looks good, and wanted to pay us part of the deposit to hold the rental until all the verifications are completed.

Do I send an acceptance letter to all 3 and whoever puts down the security deposit first? 

City of chicago has all sort of rules about security deposits.

So you might get rent and deposit before first payment due....or as you suggested might have to make 1 or 2 payments before rent starts coming in. 

He hasn't responded yet, nor deposited the rent as of 3pm on 2nd.

In regards to a deposit, I am used to longer term rentals so in a short stay like this what is typical?

I cancelled the transaction and requested they sign a release of my deposit, (which was held at my attorney).