Here's an answer for you from Europe - a third continent! Your answer from Australia says you have to say 'listening', and your answer from the US says you have to use 'listen'. I think you can use either form. If you use the gerund, you're treating the verbs as nouns, as examples of the 'things' that you can do. If you use the base form (listen), you are referring back to what you can do: you can listen or you can chat. Either form is correct.

It's okay to use the present simple tense in this sentence, because you're talking about things you can ALWAYS do there. Consider this alternative: "I like to go to my grandmother's house, because I can eat whatever I want there and stay up late." In this case, the gerund (-ing) form wouldn't work, would it? When we talk about things that are always true, we generally use the present simple form of the verb.

You need to say "listening" and "chatting" since you need to you're using the verbs as nouns (ie. as gerunds). Otherwise it's fine.