I read once that you’re not supposed to use sin/sitt/sina after och so sth like “hon och sin hund” would be wrong, and also to not start a sentence with them bc the pronoun they refer to has to be in the same sentence and precede them. Is that right?

ordlista:

Yes, all of that is correct.

You can use sin only in the object part of a sentence, not in the subject part. In this sentence you can use sin after och in the subject part, as both hund and katt are grammatical objects:

hon går med sin hund och sin katt – she walks with her dog and cat

But here both hon and hund are subjects:

hon och hennes hund går sin vanliga promenad
she and her dog go their usual walk