I am more than comfortable with words having multiple different meanings depending on their context. But now that I am starting to learn longer and longer sentences I am beginning to get slightly tripped up. I am wondering if other languages have this same problem, for example:

我家不在鄉下。

家 could mean home or family, but in the context of a sentence my brain tends to pick one before reaching the end of the sentence, ie here I read "My family not in countryside", where the intended reading is "My family doesn't live in the countryside".

Do other languages work like this as well, and does this just become more natural over time?

I feel, especially in China, that ones home also includes ones family. So saying my family doesn't live in the countryside also implies that my (family's) home isn't in the countryside.

Even if I write in English;
My family doesn't live in the countryside.
My home isn't in the countryside.
It doesn't really clarify whether I live with my family or not in the same home. So it seems just as ambiguous to me as having one character with two translations. If without a shadow of a doubt you wanted to know you would have to clarify;
My family doesn't live in the countryside, but I do.
My home isn't in the countryside and I live by myself.

If the sentence wanted to emphasis the possibility that the speakers family lived in a different location than them they might have written; 我家人不在乡下 or 我父母不在乡下.

If that confused you, then don't even try to learn Japanese.

So what you are saying is Japanese also has words that change their meanings depending on the context later in the sentence?

I think every language contains words which depend on the context of the situation or sentence. It's hard to notice of one's own language. But I think this happens even more with languages which have few syllables, like Mandarin and Japanese.