It is better to learn the verbs along with their prepositions... we often use "começar + a + verb".
is there a pattern to it? In this same exercise there are many instances of .. I think they are "modal verbs" + infinitive with no preposition.
Are there certain verbs with which you shouldn't use "a"? I've been marked wrong a few times because Duo said it was an extra word.
Some verbs use "a" some do not. As Paulenrique said, you just have to learn the verb along with whatever preposition (if any) should be used with it.
What I noticed is that at least with 'precisar de' and 'gostar de' it would be, for instance, eu gosto ver á praia or você precisa correr? So without the prepositions, why is this the case here, and not with 'começam a cair'?
You have to option to use with the verb começar: comerçar + a / começar + gerúndio (usually used with steps, routines, etc.) So, you need the prepositions.
The same is for gostar. You always use de after ir: eu gosto de ver a praia.
You are missing the verb "begin;" the word "behind" is a preposition that is the opposite of "in front of."
I wrote "fall down" instead of "fall" : how can this be considered a mistake?
To be honest, fall off would be better. I presume someone who falls down is somehow attached to the ground, but someone who falls off is not. Leaves are not attached to the ground... Maybe I'm wrong.