Is there a short way in Greek to say "at someone's home"? For example, can we say "θα βρεθούμε σε μένα", "θα πάω στον Γιώργο"
a short way to say "at someone's home"
The two examples you gave are correct.  Suppose you want to go to Alexandros’ house, you can say for short:  

  

Θα πάμε στου Αλέξανδρου = We will go to Alexandros’ (home)  

Θα πάμε στον Αλέξανδρο =  We will go to Alexandros