yes
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.
A "visa" is kind of like temporary citizenship. A "work visa" allows you to work in a country you aren't a citizen of, and a "visa" allows you to live there.
no
It isn't just for TNA, if you are not a citizen of the USA and you want to work here you have to have a Visa
You would need an appropriate work visa in that case.
She would need a UK work visa.
Yes.
Yes, unless you are already a US citizen you will need an immigrant visa that permits you to work, not a tourist visa.
Since the F-1 visa expired, they no longer have current legal status, so it is not possible to renew or get a new F-1 visa. A work or investment visa is out of the question because she would not be eligible for one due to her non-legal status. Basically, the only way to stay IN the U.S. and get legal status is by marrying a U.S. citizen.
NOPE
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 would need the appropriate work visa from the Indian government.