Porque eu preciso escrever "a nadar" aqui?
But 'nadar' alone is "to swim", isn't it? Is this sentence effectively saying "she started to to swim"?
Many Brazilians leave out the "a," so I am unsure why Duolingo is marking as the only option. Yes, it is correct...but hardly used.
Not in this case. "Ele começou a nadar" does not work. "A" here is a preposition and should be used after the verb começar!
Ela começou nadando
It's also right if you have a list of things she did in the past.
Is the "a" here "the" or "to"?
And is it always needed to say either "the" or "to" here, and if so, why?
"A" here is a preposition.
Yes, you should use "começar a + infinitive". Some verbs in Portuguese require prepositions, and it does not work the same way in English...