There are definitely times I have asked someone if the thing they have is the one i gave them as a gift so I am wondering if this sentence works that way. Probably i would know in the case of a dog, but if someone is wearing a scarf and I'm wondering if I gave them that scarf (because I vaguely remember the gift and not sure if that is the same scarf) I can see a question like this one.
No to the first one, you can't use tőlem to express "the dog is mine."
But yes, tőlem van means it's "from me" - a gift I gave, a letter I sent, something like that.
A gift from Jacob to Jan would be Jacobtól Jannak and a gift "from me to you" would be tőlem neked.
That still leaves us with the fact that this is a slightly weird sentence (presumably the speaker should know whether the dog is from him or not).