I can only assume that Charlie Brown's comment, about the dogfight, is a turn of phrase that has fallen out of favor in the 50+ years since the strip first saw print.
Question 1: Who dressed Snoopy up in that outfit? He still doesn't have an explicit owner, nor opposeable thumbs. Evidently it was someone who appreciates tartan.
Question 2: How did Schroeder know where Snoopy was going?
Snoopy's role here is subtly different from his original personality. Here, he is sort of an honorary kid. He can't talk, but Charlie Brown and Schroeder know he can understand them. The disconnect between his obvious nature (dog) and the kids' treatment of him (colleague) is what provides the joke.
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