We performed a go-around from the flare due to a rather long float.
Yes, the 172 does that if you fly the approach too fast (this is a classic complaint from someone trained on a 152 which seems to be less fussy). Fly the correct speed and it won't happen.
I thought the climb to 300ft with flap 20 was rather slow.
I was taught 200' rather than 300', in either case the idea is that you're in a stable climb by then with enough speed to keep flying with no flaps and enough height to sort yourself out if you screw up the attitude change on raising the flaps. At 20' after an unexpected unplanned go-around the inexperienced pilot could be all over the place, not a time to change the aircraft's configuration in such a way that might result in a significant pitch change if you're not careful.
Yes it does climb more slowly with 20 degrees of flap than with none. Are there aircraft for which this is not the case?