Interesting questions, but I'll do my best to enlighten you as far as I can.
Neither the Queen, nor anyone else in her family have any direct involvement in the UK government. It would be a constitutional disaster if it was found otherwise.
As for her income, well it comes from a number of sources, but the only source which could be classed as public funds are that from The Crown Estates income. This is income from coastal lands, rivers, forests, harbours etc. The Queen is paid 15% from the net income, which amounted to about £45M in 2017. About 62p / year for every person in the UK.
Other income is from her private investments, which is not published.
Also, there is income from land and property which she has inherited from previous Kings/Queens and are mostly the spoils of various feuding barons, dukes, lords and others over hundreds of years. There are two main sources and tax is paid on both :-
1. The Duchy of Lancaster : £12.5M in 2017
2. The Duchy of Cornwall : £20.8M in 2017 (This is actually in the name of the Prince of Wales, currently Prince Charles.
There are no other public payments made to the Queen and the costs and expenditure of other members of her family, repairs and maintainence of royal property etc. are all paid from this income. Although, the cost of security is mainly paid by the London Metropolitan Police.
The cost of royal weddings is also paid by the Queen, other than the wedding rings/bands, which are all made from the same nugget of Welsh Gold used by Queen Victoria in 18??.
On the subject of marriage, the only imitations are that a King, or Queen cannot be a Catholic. They can be white, black, blue, pink, or yellow, makes no difference.
As for King Edward VIII abdicating in 1936, that was a problem of his own creation when he tried to interfere with government and walk all over established protocols. It was not only the UK government who objected to him, but also the Australian, Canadian and many other Commonwealth countries. The fact that he supported the Nazi party didn't help either.
