I would not waste time with this.  Whatever check they did send you is "fake" anyway.  Tear it up.  Stop replying to them. They are scammers out to rob you.

I just got one yesterday from Kiev, same verbage

This is a DEFINITE scam -  same exact message sent to us.  I have simply reiterated our payment policy and I don't expect to hear from them again. 

I have received numerous scams that are similar. Generally they want to rent for 2 weeks or more. Often it is an alleged group of doctors, ministers, researchers or group of people on their honeymoon. They all want to know the same consistent set of checklist information about me. They usually want to write a check for a greater amount than the rental and have me forward a portion to someone else for a rental car or some other thing.  The first time I received one I researched the persons name and found the scam is to send you a fake check, have you forward part of it to a 3rd party immediately after you deposit their fake check. You are out whatever you send to the 3rd party because the check was fake and hadn't cleared the bank. Before sending back any money or paying for anything on behalf of anyone, contact your bank and confirm the check has cleared, in writing.

Has anybody received an inquiry from XXX or XXX?  Jason is most aggessive - asking to rent all of February - didn't ask price, but requested that I send all my contact information so he could send payment.  After asking what his purpose for visiting was - he responded - "official business" .... red flag - I called the number he listed - fax machine or leave a message ... I asked him to call me or to email me a number where he can be reached and no response yet.  what do you think?

Sounds questionable.

This is the exact wording of the one i got, just different names.  I wonder if the people are even from overseas.  I have a suspension that they are actually American, but the scheme wouldn't work as well.

I called them on the scheme as soon as I received the email like yours below and never received a check.  My first thought was they were trying to launder money.