There's other ways to get the peracetic to form faster--you can raise the temperature and reduce the time as in the document that Rhodium mentions. Azeotropic removal of water with ethyl acetate (or whatever) doesn't look promising to increase the concentration of peracetic acid much--perhaps I'm mistaken on that, I only briefly reviewed the journal. There should not be much oxygen coming from the peracetic... an occassional tiny bubble can be seen, but it's very intermittent.
Yes, running the peracetic with a cold water bath, and just letting it to come to room temperature is the way that I did it. The references I read recommended keeping it cool--I found it important at the beginning if I didn't want the flask to hit a reflux.
