Q: Is the primary domain controller supposed to be the only domain controller in my DefaultFirstSiteName site that receives replications from it's sister site?
A: I think what you're really asking here is whether or not you should have inbound replication connections from DC3 to DC1 and DC2, and the answer is no. One aspect of the job of the KCC and the ISTG is to create a least-cost, loop free replication topology. If both DC's in the DefaultFirstSiteName site had inbound replication connections from the DC in the sister site then a loop would exist.
Let's assume DC1, DC2, and DC3. DC1 and DC2 are in the DefaultFirstSiteName site and DC3 is in the sister site. DC1 has an inbound replication connection from DC2, and DC2 has an inbound replication connection from DC1, so any change made on either of these two DC's is replicated to the other DC. DC3 has an inbound replication connection from either DC1 or DC2, but not both. Likewise DC1 or DC2, but not both, has an inbound replication connection from DC3. If DC1 and DC2 both had inbound replication connections from DC3 then a loop would exist. If DC3 had an inbound replication from both DC1 and DC2 then a loop would exist. What would happen to a change that occurred on any of the DC's in this scenario? The change would "collide" as it replicated from DC to DC.
There is a lot at play in AD replication and in building the replication topology but in answering I tried to keep it simple.