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.
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.
US and UK allow Dual citizenship. A person can be a US citizen and a citizen of UK if he/she desires to. But if a person applies to get naturalized as a citizen of UK when he/she is already a US citizen, then it will lead to losing US citizenship.So a person can have US & UK citizenship as long he/she is not a naturalized citizen of UK.
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.
In the way of actual paperwork, none apart from the marriage license. The UK citizen comes in on a visa waiver and you can get married in the US. For the US citizen to get to the UK, that is where the paperwork comes in. You both need to go to the UK embassy in the US and apply for a marriage visa for the US citizen to come into the UK. The US spouse will then be able to work straight away when you get to the UK also based on that visa.
Get a working visa, and use it. Apply for and get permanent residency.
A US Citizen would need a work permit to live and work permanently in the UK.
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.
No, you will have to get divorced, you will almost certainly get sent to prison for bigamy.
Get court approval
The UK citizen must obtain permanent residence.