Amateur Rambler / Proud Ergi
I seem to remember that Puhvel in his Comparative mythology claimed an old Indo-European god showed up as Odin, Apollo, and Rudra (the Vedic precursor of Śiva).
Curing and causing (with arrows) disease: Apollo and Rudra.
Wearing a hat: Rudra and Odin.
Non-standard number of eyes: Rudra and Odin.
Prophecy: Apollo and Odin.
Mice or rats (or moles?): all three.
At the same time, I'm not sure of a Proto-Indo-European God has been reconstructed from these three figures. I think part of the problem is that, unlike other PIE Gods, the names we call them now probably started as bynames or titles, rather than something we can use linguistic reconstruction to point to.