The context: They were talking about a gate they couldn't open and pass through. But they weren't so eager to venture too deep into the forest to begin with.
... но мы не особо-то сюда и рвались.
I tend to use "рваться" with an infinitive or а preposition "к/в", so I'd be tempted to place an additional verb between "сюда" and "рваться".
Is this how native speakers commonly express the idea? And is there any other instance where the form "рваться + adverb" is used?
For the purposes of modern usage and disregarding the etymologies сюда is a Locative case form of pronoun тут/здесь and as such already includes prepositions of direction, which hence become redundant. Apropos hence, an English equivalent which doesn't require prepositions, along with thence.
Neither infinitive is necessary. Expressions such as рваться домой / рваться обратно / вперёд / назад are fully idiomatic.
A higher register synonym of рваться in this sense is стремиться, which nevertheless isn't always a viable alternative.
It's used with both prepositions and adverbs. Quite often with prepositions in sports, fighting, warfare, competition contexts. Cf, рвётся в бой (in a fight), рвётся к воротам (in football) - sometimes literal, sometimes metaphorical.
Since this word implies somewhat intensive emotion (long, want, yearn) it MAY suggest lower register sometimes, depending on the context. But may express some strong feeling in neutral or higher styles.
One should just be more cautious with any words denoting strong, intenstive feelings, this word being no exception.
And is there any other instance where the form "рваться + adverb" is used?This is called "an adverb of direction" (наречие направления). There are quite a few of them.