Agreed, fungal strands/mycelia. - Mycelia of most kinds of fungi look same/similar, so identification of species is unlikely without microscopy, or in some cases DNA. - Habitat narrows down the possibilities as there are fungi that are more keen on living on/in houses. - If it produces a fruiting body (mushroom) that could make identification by photo possible e.g. Peziza domiciliana or Pleurotus sp. -Generally (given that few fungi with that habitat/substrate are harmful) it is unlikely to be harmful to your puppy, but certainty would require identification. Hope that is at least somewhat helpful.
Here's what grows out of a stump on the outside of the house (close to the house next to the crawlspace)
Unlikely to be the same species as the crawlspace fungi, partly re reasonably habitat limitation guesses, and partly as there are so many fungi species around just waiting to fruit, and you can have very different kinds coming up next to each other. Not sure what your stump mushrooms are, at least not without more details, but overall they do have a sort of Armillaria sp. (honey mushroom genus) look to them.
re the crawlspace mycelia: @Sulev - It is possible but hard to say without the characteristic fruiting body. I expect rhizomorph appearance would be recognizable to someone more familiar with it than I ... but in general the mycelia in the photo don't look robust enough. Also if @nifter is from BC it would be less likely, as far as general range of that species.
The Serpula lacrymans should be quite reliably identifiable by just looking this mycel with a microscope. Of course a fruiting body or spores can make identification more easy and with higher certainty. Once I took a sample of very similar substance (mycel) from the house of my relative to the laboratory for ID, and it appeared to be a Serpula lacrymans. There are not so many fungi, that have so aggressive growth in relatively dry environment.