I am on holiday and we are renting a house for a week which owns a big apple tree and we were wondering if we can pick the apples to eat. A lot of the apples are also beginning to rot if not picked. So can we pick the apples?

You can pick the apples which are beginning to rot (with accordance to Shulacha Aruch Choshen Mishpat 260:6. Sema 23). You should not pick the other apples, unless specified in a contract or advertisement of this rental that it includes picking apples from the apple tree. Best is to call the landlord and ask him for permission to pick the apples.