I have been through a very similar situation. I would be cautious as if they are rushing you they may be trying to hide information from you. One thing I would spend a lot of time looking at is other employees that they have/had. Look at there work history and talk to them if you can.

From my experience (lesson learned the hard way), one thing to be 100% confident on is that what you are signing, the employment agreement, is solid. Pay the money for a good lawyer to look it over.

Let me know if you need a reference for a lawyer.

Also them saying, "who ever says 'yes' first gets the job" would make me very suspicious as they are not valuing your skills.