what ticket do I buy for my 5 day stay in HK Island?

You are suggested to buy an Airport Express Travel Pass, which can be used as an Octopus card when topped up.
As for Q2, you can take Bus E21 and transfer to Bus 102 at Cheung Sha Wan Station, Cheung Sha Wan Rd. And then get off at North Point Government Primary School, King’s Rd. After that, walk south for 3 minutes, you will get to Quarry bay.