A US trained doctor can work in the UK with the proper visa. There are many jobs available in both the US and in the UK for medical professionals.
No - a British citizen is automatically entitled to work in the UK
She would need a UK work visa.
A US citizen would have to apply for a visa and become resident. Alternatively they could marry a UK citizen. In addition a British Company could invite the US citizen to work in the UK and apply for a work visa on their behalf. There are quite a few ways to do it legally.
You must be living in the UK and have a right to work there. As a non-UK citizen, this will mean you will have to have a visa that allows you to work in the country or you must already be a citizen of an EEA country.
Whether a US citizen can work in Japan is completely unrelated to that person's UK residency status. It is a matter for Japanese law and regulations.
Get a working visa, and use it. Apply for and get permanent residency.
only if they are straight
You don't have to be a UK citizen to marry a UK citizen. However, you won't automatically be given the right to live in the UK just because you have married a UK citizen.
No, you will have to get divorced, you will almost certainly get sent to prison for bigamy.
A US Citizen would need a work permit to live and work permanently in the UK.
Just a work visa
If you're not a British Citizen, you will need to apply for a work permit first.