I would expect that in both cases an arp request will be sent, and for the first the real host will answer and for the second the qualifying router will answer.
No, the router only provides an ARP response if the ARP request is for the IP-addresses of one of the router's own interfaces.
The client uses the netmask to decide if the destination IP is LAN-local or needs to be routed.
If the client determines that the destination is outside the LAN, it selects an appropriate router from its routing table. Often this will be a "default gateway" configured statically or, more likely nowadays, by DHCP, at startup.
The client would then broadcast an ARP request to find the MAC-address for the router's IP address, after receiving the arp response it can then send packets to the router that have the router's MAC address and the final destination's IP-address. The client also stashes the router's MAC address and IP-address in its ARP-cache to save having to do further ARP requests fro the same info.