If a router receives an ARP request in which the destination hardware address is a broadcast address, and if the destination IP address is an address that the router cannot find in its routing table. What will the router do in that situation? Is it going to cache pair IP/hardware source address and send ICMP message "Destination network unreachable" back to the source?
ARP is a layer-2 thing -- it's how layer-2 address (MAC) is associated with a layer-3 address (IPv4). The router (any node) will only respond to ARP requests for addresses it owns -- the addresses assigned to it's interface(s) or aliases. It will silently ignore all others.
If your router is setup to handle the ancient bad idea of proxy-arp, then yes, it will respond to any request for which it is capable of reaching. So if there's a route to the requested address -- include a default route, it will answer with it's MAC, otherwise the request is ignored. If you have two proxy-arp'ing routers, one with a route, one without, both will see the request, but only the one with a route will answer.
If nothing answers the ARP request, the requesting node itself will generate a "host unreachable" error.
What does a router do with a packet if it can't find the destination IP address in its routing table?
If a router has no route to the destination IP, it drops the packet and returns an ICMP Destination unreachable message to the source IP.
If a router receives an ARP request in which the destination hardware address is a broadcast address, and if the destination IP address is an address that the router cannot find in its routing table.
The router only anwers ARP requests for one of its own IP addresses. All other requests are ignored. Whether is knows a route to the ARPed IP address or not doesn't matter. (As @Ricky has already explained very well, including the obsolete proxy-ARP bit.)