I think even for the second one slightly changes what is probably meant. Anyway and also are the best bets.
However, they would not necessarily be the most usual way of expressing the sentences in my neck of the woods. That would be:
My neighbours are very noisy, but we still get along very well.
You will need to clean the bedroom every other day, and the bathroom as well.
Other ways of expressing this:
My neighbours are very noisy, but all the same/even so we get along very well.
You will need to clean up the bedroom every other day and the bathroom, too.