The HAP portion of the rent comes in by ACH on or before the 3rd of the month.The tenants themselves need to be screened properly like anyone else.  

Give them an option to pay online or by ACH, for instance.  

Do I understand correctly that I erentpayments.com allows ach debit allowing the landlord total control (tenant can not cancel the auto deduct from their account without landlord approval)?

Bank to Bank ACH transfer.

That way the head ache of evicting is in the sellers hands.

I have a lead of another student who is interested in renting a room who is in the same program as I am (medicine). 

You could encourage her to switch over to automated ACH payments with a discount perhaps.  

I'm in the approval process to accept ACH payments. 

We typically only deal with ACH but I would assume there would be similar rules for regular checks also regarding the time frame to return them.  

I'd suggest agreeing to waive the fee if they elect to pay automatically though ACH transactions.