There are a few diseases that started as cancer: https://en.wikipedia.org/wiki/Devil_facial_tumour_disease

Generally, infectious cancer is a result from a population that either regularly has open wound contact, and or is very closely related genetically.

Given enough time cancer can even start to survive on it's own: https://en.wikipedia.org/wiki/HeLa It's actually become somewhat difficult to avoid cross contaminating samples with this cell line.

There's at least a couple mechanisms. The most obvious is heredity—certain alleles of a particular gene that normally controls growth, or expresses some surface protein, mutate, leaving descendants with a greater risk of cancer. BRCA4, for example, has some variations that all but guarantee breast cancer. Mutation can happen in both somatic (body) cells and in germ-line cells; when it happens in the latter, the mutation can be passed on to offspring.

Oncoviruses are another transmission vector. HPV, or human papillomavirus, is a virus that causes warts (of all kinds) in humans, and also causes several kinds of cancer, including cervical and esophageal cancers.

A third way would be a pretty rare thing in humans, but is thought to be behind some of the canine cancers you mentioned: direct transplantation of cells. The tasmanian devil is on the edge of extinction thanks to a facial cancer they get from each other through this mechanism.

You are right, I was referring to the fact that cancer cells are rewarded as being "more fit" in the body than their neighbors. By gorging themselves on resources and disabling the body's mechanisms for slowing growth, they are rewarded with proliferation.