answer is yes or no...
For the very first packet it will broadcast arp request for the destination ip address and default geteway ( i e - router ) will reply with his ethernet interface mac-address as l2 destination address. then pc1 store this info into cache memory and build packet with PC2 ip address as L3 destination and router ethernet interface mac address as L2 destination , Once the packet reach at gateway router , router will check it's routing table and next-hope to forward that packet , it will again build packet with P2 ip address as L3 destination and Next-hope interface mac-address as L2 destination .... and packet will proccess at each hope in same manner untill it reach at it's destination ... then destination machine i e PC2 revert with PC1 ip address as L3 address and his gateway mac-address as L2 destination address..
During first communication all the devices store ip-mac binding in cache memory hence rest of the packets for this session will bypass arp resolution and send directly by info stored in cache memory.
The only case that pc1 would arp for pc2 directly is--
1) they are on the same subnet (determined by the local IP + netmask compared to the destination IP--pc2)
2) if the pc thinks they are on the same IP subnet (incorrect mask)
3) there is no route (default gw is not set on pc2)
The arp in 2 or 3 is considered a proxy arp. In that case, the first hop router would respond (unless it has proxy arp disabled)
As for calling something an Ethernet network or internetwork--stuff like this, I'd just always consider the context. Strictly speaking, it may be an internetwork that is comprised of multiple Ethernet networks.