Sure, Peter is a comic character. But keep in mind that the comic version could have more complex plotlines and interactions with other characters that might not have been fully explored in the movie.
Peter would probably be shocked at first. I mean, getting a call from Deadpool in the middle of class is not an everyday occurrence. He might quickly look around to see if anyone noticed his phone ringing.
I'm not sure specifically which fanfiction you're referring to. There could be various interpretations. Maybe it's a humorous take where Deadpool stumbles upon a young or miniature version of Peter and gets into all sorts of wacky adventures with him.
Yes, there are quite a few. You can start by checking fanfiction websites like Archive of Our Own. There you'll find various takes on the 'deadpool chibi pet peter' concept. Some might be humorous, while others could be more action - packed or emotional.