because they could be carrying a bomb
A 'green card' grants you official immigrant status in the United States. It has nothing to do with the UK.
No You have to realize that the UK is not in the Shengen Area, which means you cannot go to the UK with a Schengen visa. You will also need to apply for a UK visa separately for the UK, for example, you may need to apply for a Standard Visitor Visa if you plan to travel to the UK. Visa requirements differ with each country, even within Europe, as there are multiple regional agreements and different types of visas, so check before you start making travel plans as not every country will allow a holder of a specific visa entry.
Yes, a US citizen can work remotely in the UK without needing a work visa as long as they are not physically present in the UK for more than 90 days in a 180-day period.
No, a UK visa is only valid for the UK. You need a "Schengen visa" for Switzerland.
South Africans require a visa to enter the UK, but it is not a difficult process to get one; just start the process in good time. You may not know that South Africans now require a visa just to transit through a UK airport.
Are you seeking a visitor visa for the UK?
No need for a visa, as the UK is a participant in the Visa Waiver Program.
I once traveled to Switzerland on my UK visa.
No. Northern Ireland is part of the UK, so the visa will cover it.
At this moment in time Botswana Passport holders do not need a Visa to enter the UK. However, this may be subject to change in the near future as South African Nationals have recently been included on the list of nationals needing Visas to enter the UK (March, 2009) and Botswana's requirements are due to be reassessed in September, 2009.
No, individuals with a Schengen visa cannot enter the UK. The UK is not part of the Schengen Area, so a separate visa is required to enter the UK.
You can establish your own business in the UK if you have the UK work visa.