Wow. It's not really valid to imply a connection between a cooperation or mirror test to Theory of Mind, though I can see where they're coming from, but trying to connect Theory of Mind to spiritual beliefs is a stretch only Mr. Fantastic could do.
An example of an experiment testing Theory of Mind
is this. You can scroll down to the appendix which contains little story snippets and questions to assess different levels of Theory of Mind. For example, in the first question, it is stated that Bobby keeps his chocolate in the cupboard, but when he's away, his mother takes it and puts it in her bag. The question is, when Bobby returns, where does he look for the chocolate?
I recall that they tested this in chimpanzees by putting one in a room with two people, one of whom the chimp saw hiding food in a location inaccessible to it, and another person who did not see this. They saw that the chimp would beg for food with the person who knows about the hidden food and ignore the other person, which suggests it has Theory of Mind. That says precisely diddly squat about whether chimps can have religious beliefs, though.