1) Is it ok if a cat walked over a prayer rug?
2) Can one have dog as pet in backyard with their own dog house and own everything and not let them enter the house?
3) Was Habashi the killer of Hazret Hamze, (a) forgiven. By the prophet? & did he became a Muslim?

1) Yes. It is OK, as long its feet are clean (Taher).
2) No problem, but not recommended.
3) This what is narrated in the history books