In what way do you mean recover, I would expect them to be fine after the walking that you would do to finish off any work session, if still blowing after 5 mins walking then they are either not fit enough or in discomfort, it is not comparable to galloping or xc where they are expected to blow for a few minutes however fit they are as they will be working to near full capacity, the same for heart rate a pessoa session should not be unduly stressful and the winding down few minutes should allow heart rate to return to near normal.
Depends on many factors. My PSSM mare takes a very long time to cool down after even very light work. My sports horse could gallop 4 miles and barely be blowing. The mare is essentially 'fitter' in terms of how long she has been in work and how often she is ridden, but you wouldn't think it.
Working in a Pessoa is pretty hard work for even a fit horse so it depends on fitness levels, what setting you have the Pessoa on, how much walk/trot/cantering you're doing. My fit eventer never goes in the Pessoa for longer than 20 mins gradually increasing to the highest setting and mainly doing trot work. By the end he's pretty nackered and we then go for an in hand hack around the yard for 5 mins and he's recovered. By recovered I mean no longer still sweating and breathing has returned to normal.