Keep those lines on communication open.Check to see what limits you can have for a security deposit. 

Don't order the title search till you've gotten an earnest money deposit put into escrow.

When submitting multiple cash offers with no contingencies and a large earnest money deposit, how do you guard yourself against having multiple offers accepted but you only have the cash to close on one of the deals (assume you received two signed contracts the same day)?

Place rental ads around the neighborhood and onlineFind renters while continuing collegeScreen renters and have them fill out deposit agreements, and other contractsMake renters pay the first of every month and make them pay for electricity.After college rent all units outRefinance after moving outEach 5 years try to refinanceMove on to other propertiesAfter 30 years start the selling processPlans for turning a profit on first dealHave 10% worth of reserve fundsBuy using an FHA 203b Loan with 3.5 % downRent other units not occupied by meRefinance whenever possible for positive outcomes of loan termsRefinance for cash out for other deals.Move out of unit and rent that after 4+ years of collegeMove on to other deals while this continues to thriveWait 10+ years until home is free and clear and appreciation is high unless the local area turns below my requirements before 10 + years and then sellSellIn summary of profit plans, the main way for my gains is through appreciation and cash flow and taking out equity if possible.Team ComponentsYour Mentor - Every successful entrepreneur needs a good mentor - a guide.

If the tenants don't quite meet your qualifications, you might consider requiring a larger security deposit or a last month's rent for additional security.

Terms of sale - $2000 deposit, 25% of bid due in 30 days and full balance due within 6 months with unpaid balance at 12% APR.

If they have a unique rent amount, I give them a choice of paying online via www.erentpayment.com, or depositing it directly into my business account at the bank.

Can I just oblige by her request and take their security deposit for that lost month (Septemeber)?

This is not a deposit but a fee for simply having the luxury of a pet at the home.  

Its equal to the security deposit.