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Color keying is not going to be much help here since, as you note, the dog has the same color as the background.
Depending on how precise you need the resulting matte to be, you have many options.
Rotoscoping is the extremely tedious way that'll give you the best results.
Using the Roto Brush tool will be much less tedious, but the results won't be quite as good as carefully done manual rotoscoping. Of course, you could get a pretty good first pass with one technique and then touch up with the other. (Heck, even color keying work normally involves some rotoscoping touchup.)
That's what I was afraid of. Stupid question - how do I access the matte generated by any of the key effects so that I can paint on it? Sorry for my ignorance - I spend most of my time in Photoshop.
If you're so new to After Effects, I recommend taking a step back and working through the getting-started materials first:
Diving right into a complex project with After Effects is not generally successful or pleasant.
That said, here's a page with an overview of ways to interact with transparency:
Note that one of the ways mentioned on that page is painting on the alpha channel. There's a link on that page that you can follow for more information.
Looks good - Thanks!