Is Windows 2012 failover DHCP (hot standby) compatible with Windows 2003?
I'm not sure if it's enough to create a Windows 2012 DHCP server and configure failover options.
No, this will not work. Both DHCP Servers must be at least Windows Server 2012.
As an aside, you should be removing 2003 from your environment quickly, not tying it in to new infrastructure. It goes end of extended support early next year.