I feel like the rule for these competitions is ‘keep going until you get tired of getting hit in the head’

Get the opposing team members on the ground. Once down, they're out of the fight. Best I can tell, that's the only rule.

"The field is mine, for you have been tripped, Sir Reginald! Thoust spends more time on your back than thine whore of a mother!"

Armor works.

My understand is - please correct me if I'm wrong - neither. HEMA Alliance and SCA do their own thing, is my understanding, while 'Battle of Nations' is organized by Buhurt Technologies, which produced the armor for this event.

BOTN is one type of "buhurt" or "bohurt" which is very much it's own thing and neither HEMA nor SCA (though there are people that do two or more of these activities)