Can a company hire you, give a start date and ask you to put in your 2 weeks notice then turn you away later because of an unsatisfactory on your background check?

The company probably can do this.  There are no hard and fast rules about offering someone a job.  The offer is usually contingent on passing a background check.   Since you did not pass the background check, the company is not obligated to hire you.

I am sorry this happened.  I hope you can get your old job or a new job right away.