Yes, a US citizen can travel to Germany.
Yes, a US citizen can work in Germany with the appropriate visa and work permit.
Yes, but the American citizen must first apply for a fiancé/fiancee visa on the German citizen's behalf. This would be the I-129F petition. It would have to be approved before you can marry. This is if you plan to remain in the USA afterwards. For more info check out www.uscis.gov
Yes they can live together
Yes, a US citizen does not need a visa to travel to Germany for stays of up to 90 days for tourism or business purposes.
Australian female citizen marrying US male citizen - which would be easiest marrying in US and come back to Australia to live or marry in Australia to live?
15 YEARS
Yes, but the US citizen should sponsor his/her spouse's immigration in order for him/her to get a green card.
Permanent resident can apply for US citizen after live in US for 5 years (continuously and not leave the US in the period of 5 years).
Not unless they become a citizen of the USA - you can't have a person who isn't a citizen of the country run the country. That just doesn't make sense. So you have to live in the USA to be the president of the USA. If you live in Germany and want to be president, try running for the president of Germany. You would probably have a way higher chance of achieving that before you became the USA president.
Yes
You can but be aware, that the UK citizen cannot legally change their status while in the US. It depends on where you both want to live. If the US citizen wants to come back to the UK, a marriage visa is needed from the British Embassy in the US. However if the UK citizen wants to live in the US, they will have to go back to the UK and the US citizen will then need to file the I-130. Until it comes through you cannot live in the US, but will have to wait in the UK.