If you go on the Opera house web site it will give you all the shows that are currently on. If you are staying in a hotel the reception or concierge will be able to help you.

No dress code is a rule but too many people went there casual

You should be able to do all that in the morning. There are tours of the opera house you might not have time for if you want to wander around the gardens longer - but they are all right in the same place, so if you have an entire morning you will be able to get a good feel for all three.

April in Sydney is Autumn : it's warm but not as hot as in the summer months. Very nice for walking about. Expect daytime temperatures around 20°C (68°F) cooling down at night to 15°C (58°F) .

you do not have to pay to see the opera house from the outside. if you would like to see a performance you would book on line and there is a cost its lovely to walk around the opera house have drink and a meal at one of the bars that surround the Opera house lovely day and night

The Opera House plaza abuts the Botannical Gardens. The plaza is large with multiple levels. From the Opera House building itself the entrance to the Botannical Gardens is less than a five minute walk.