Australian traveling from US to Cuba. Will Trump's new rules affect me?

YB - exactly.

(And I suspect that in the US they check that the two match ! In Panama once, the airline check-in gave me someone else's boarding pass, and neither I, nor the main passport control noticed.)

At every stop along the way until you enter the plane they check to make sure both boarding pass and passport are a match.

#8 there are NO US exit checks, there will be people from the airline and probably the TSA checking that your passport matches your boarding pass but there is no immigration or CBP check when you exit the US. A lot of countries have done away with exit checks as they figure they get more bang their buck by concentrating on entry checks. How do you think all the illegal immigrants in the US can leave without any problems if they chose to do so?

Looks like exit checks are coming and being tested,

http://www.vocativ.com/435077/jetblue-face-recognition-boarding/

http://viewfromthewing.boardingarea.com/2017/01/28/new-executive-order-demands-immigration-checks-leave-us-not-just-arrive/

#14 the US government has a budget to spend on border checks, whether entry or exit checks, for all the talk both during the Obama or Trump eras they pretty soon figure out they get more bang for their buck by spending their budget on entry checks. Sure if they could find a solution to exit checks that does not require any money they would jump at it but I think that is just pie in the sky right now.

The Trump era is only 6 months old and he ran on an anti immigrant tight borders campaign.Do you know more about this than Trump has let on to?

If you have an Australian passport and are not a USA resident, Trump has no control or jurisdiction over foreign citizens of other countries than the USA. Nor does any American president. They are not world dictators. Some, like Trump, think they are world dictators.

How do you know dual citizenship laws of both countries?

The US rules apply to US citizens wherever they are resident and everybody whilst they are physically in the US.

I think every country would claim jurisdiction over foreigners whilst they are in that country.