How does a DHCP client communicate with a DHCP server when it is first started?
By broadcasting and receiving broadcasted datagrams.
What is required to enable a DHCP client on one network to lease an IP address from a DHCP server on another network?
A DHCP relay agent.
Can a router be a relay agent? Can any router be a relay agent?
Yes. A router can be a relay agent. No. Not all routers can be relay agents, only routers that are RFC 1542-compliant.