Yes with the following caveat: "For the comfort of Executive Floor guests, children are not encouraged in the Executive Lounge when alcohol is being served."

Buses: http://www.sydneybuses.info/
Trains: http://www.cityrail.info/
Ferries: http://www.sydneyferries.info/wharves-and-maps.htm

Your best bet is a weekly MyMulti1 for $41 AUD which covers Train, Rail and Ferries with the exception of to/from the airport which is an extra $3.95 AUD gate fee each way. More info and answers to additional questions about Sydney Transport may be found at the links provided above.

1) Botanical Gardens - leave hotel and walk toward Sydney Opera House to gain entrance to Botanical Gardens Opera House entrance.
2) The Rocks - out hotel, turn right and area between hotel and Harbour Bridge is the Rocks.
3) Walk/jog the Harbour Bridge for free (not the Bridge Climb for $199 AUD) - walk to the Rocks and take a left on Argyle Street up the hill to stairs on the right up to the Cumberland street, then across Cumberland street to another set of stairs up to the bridge. On the other side of the bridge is Luna Park, Swim Stadium, Milson's Point, Kiribilli where you will find Admiralty House (Governor General's residence) and Kiribilli House (Prime Ministers residence when in Sydney).