I was thinking along the lines of a $100 Amazon gift card to apologize for the role I played in his week of inconvenience.

It was an interesting lot, the county created a handful of parcels on the lake and opened it up as a lottery to win the right to choose and buy a parcel.

If I push them to get savings accounts/debit cards, are there ways to process rent with a debit card?

Heck you could likely do credit card stacking in your personal name and get 45k or so.

RE as a community we always talk about the wins.

My pest control guy hands them his card for future treatments they can pay for.

I've submitted offers for clients that are something akin to what you offered and we still lost as the other buyer just offered more money than what my client was willing to spend.Have your agent find out what the winning offer was and hopefully you can get some clarity on what it will take to win the bidding war.

These students will probably fight and win even though I took the necessary steps prior to them moving in.

You have the winning hand here.As a landlord you have to not react to the "I have an attorney." 

Wow, this can be a can of worms, but it is he/she that brings solutions to problems that wins the deal and makes profit doing so.