Can someone please explain why it's not "tør du å gjøre det" instead of "tør du gjøre det"?
Because a sentence can only have one conjugated verb, in this case 'tore' in the present tense giving 'tør'. And because 'tør' is a modal verb here it is not the main verb of the sentence but 'å gjøre' is. The main verb of a sentence when following a modal is always in the infinitive form without 'å' in front of it.
You can think of it that modal verbs are conjugated and this conjugation is shared with the main verb which is in its infinitive form.
Yep! "doing" is either a gerund or part of the present continuous tense. We can cross out the option of it being the present continuous since that is always constructed with a "to be" verb, then whatever verb that person is doing. That isn't the case here. Is it a gerund? Not really, as a gerund is a verb with -ing that is being used as a noun. You could say "I like doing it" since it is then a noun, but that doesn't really work for any of the other modal verbs - "dare" included. So yes, it is a mistake. :)
The infinitive marker is optional with "tør", as is the case with a few other "semi-modals" such as "trenger" and "behøver".
Even when you feel sure about something as a native speaker it's a good idea to do a quick Google search. Often what we feel is absolutely indisputable can be dialect or context dependent.
They can often be used interchangeably, but "å tørre" is the opposite of being too afraid to do something, while "å våge" is often more about braving the odds or daring to put something at risk or at stake in order for a chance at something else. As such, it can sometimes be translated as "to venture".
"Den som intet våger, intet vinner." (You have to play to win)
"Hun våget sitt liv for å redde en annen." (She put her life at stake to save someone else)
"Å våge" is also a good choice when someone is actually daring you to do something, or telling you what the consequences are should you dare to do something.