So, do I understand correctly that a different set of rules apply for touch and go training flights?
Depends what you mean by the word "rules".
Firstly, you're only legally obliged to obey the performance criteria on a revenue flight (which crew training is not).
On a revenue flight if you advance the thrust levers and get a take off configuration warning then the take should be rejected and you would proceed in accordance with company rules. Either way the take off flap setting should not be changed during the take off roll.
However, with respect to flaps, the TOWC will only sound if flaps are not in the take off range (for the B737 1 thru to 25) - I'd have to check the manuals but I think it's the selected setting rather than the actual but no doubt someone here will correct me on that if wrong!
A training flight is one that is operated in accordance with special procedures as laid down by the Company operations manual (including obviously advice from the manufacturer) - you also have a training captain in charge and more often then not a Safety Pilot in the jumpseat to assist and monitor.