If a machine sends a UDP packet to some destination over the internet, and that destination sends a packet to the source address in a protocol that is not UDP, will the router forward that packet to the machine or drop it?
A "router" always forwards a packet when it can, ie. has a routing entry to apply. It doesn't care about the L4 protocol used nor its state.
It depends on what your understanding of "router" is. A simple router just forwards but with firewall rules, NAT and other functions applied, forwarding may or may not occur.