If I accept the suggested pricing opportunities it could drop my rates by $100 / night in some cases. 

Check local law, whether you have to accept varies by state.

All things being equal, I would accept these applicants, but I would require them to pay a deposit equal to two months of rent.

The only slight red flag was that they only accepted payment via Venmo or wire transfer, but that's how I accepted payment when I was selling lead bundles, so I still thought it was worth a shot.

You will want to consult with a lawyer.In regards to your last two questions here's some background on what makes a contract valid and enforceable:Offer: A promise to do or not do something within a certain time period Acceptance: A clear and direct statement that all terms and responsibilities are accepted Consideration: A legal and adequate inducement given in exchange for the promise Capacity: The parties to the contract must have the legal capacity and competency to contract Awareness: A party must be aware of the contract Legality: The contract must be legal in the jurisdiction it will be operating in***Not legal advice*** Steps I would take: -Review contract in detail to understand each counterparties responsibilities (if you are not aware)- Email a copy of the lease to each of the tenants listed on the lease and reiterate lease expectations for termination notice, rent payment etc. - If the lease was signed via Docusign (other esign may have same ability) - you can go to the envelope, go to history and view the IP addresses for each action taken.

I think there is even a contingency in place for those whom submit deals that DO NOT get accepted for 3 months in a row.

If I were you, I'd pass on this property and look for something else (and I suggest a 3 BR) under $80 that you can put a little rehab into, unless insurance pays your family member a bunch of money and they accept a much lower price. 

And do they accept counter-bids after initial bid is placed?

.* No calls accepted to review receipts: I have asked Jenny several times for the last 3+ weeks to get on a call to review the receipts and all of the items that are in the house, I have given her several deadlines to resolve the issue, she has never accepted any of my invitations to join any calls despite me pleading her to do so as my credit card payment is now due and has a 29.99% interest rate if not paid.I have found several people that were also affected with just 2 facebook posts.

Some hosts also accept bank transfers, but you’ll want to ensure payments clear well in advance of their stay.