Hi, as far as I know, the 2nd is correct. And it'd be better to say: I wouldn't worry about their causing any trouble. About is a preposition and should be followed by a noun or a gerund. Please refer to the usage of gerund in your grammar book. Cheers!

As Remy Beijing points out, the grammatically correct form is actually 'their'. He is absolutely right when he says that the preposition 'about' should be followed by a noun or gerund. The gerund here is 'causing', so it needs to have a possessive 'their' before it. However, this form is rarely used, other than in formal contexts. Of the two options you've given, the first is wrong. The second - 'them' - would be generally seen as correct, as this is what almost all native speakers would say. This is a classic example of a foreign learner who knows English grammar better than most native speakers do!

The second sentence. I focus more on writing and other creative work but I'm sure that someone here can help you with grammar.