I use eRentpayment.Pro's:No paperwork or depositing checks. 

Finally digging through the dol.gov, found a helpful link: http://www.dol.wa.gov/business/realestate/assistants.htmlI see that you don't need a license if you are:Property management assistantsUnlicensed assistants who provide property management brokerage services for the owner, designated broker, or managing broker may perform the following tasks and duties:Unlicensed assistants may… Deliver lease applications, leases, or any amendments.Receive lease applications, leases, amendments, security deposits, rental payments, or any related payment for delivery to, and made payable to, the owner or real estate firm.Show rental units.Execute leases or rental agreements, rental payments, or any related payment for delivery to, and made payable to, the owner or real estate firm.Act as a scrivener, completing blanks on preprinted and preapproved rental agreements with the exact language provided by the property owner, designated broker, or managing broker.

I keep a cheap receipt book in the truck if someone gives me cash for something like a deposit.  

On deposit about two weeks after listing.So sometimes it might make sense, especially on a limited budget, to check these out. 

I presume you still hold a security deposit so you could still punish them if you'd like but what if you didn't and left on good terms?

I just want to see what each person paid, (rents all the same in each house, though I they have given me different amounts in the 2 months they have been there), what they should have paid (students), what they owe, what their security deposit was. 

Do you have it in your lease that accumulated late fees can be taken out of the security deposit? 

require a sizable deposit.

My wife spends about 15 to 20 hours a week taking payments, paying bills, making deposits, and doing the accounting. 

You'd need to know what kind of lease, how far behind, is there a security deposit, what claims the is tenant making, if any.