Is my w-2 info supposed to show my employers name?

Of course the W-2 would have the employer's name and address along with the EIN.  They are the ones that issue the W-2.  
The EIN is in box b of the W-2.  The employer's name and address is in box c of the W-2.

Yes, your W-2 form shows your employers name. 

See attachment below for direction where to find it, and for additional information about W-2 form follow the link