There are laws around elections and referenda: the various Representation of the People Acts cover most of them, though there are others.
The body that oversees elections is the Electoral Commission, which has a statutory duty to give advice and set up rules for elections, and may impose civil financial penalties (not, legally speaking, "fines" though ...
You can't "opt-out" of the full register. You can decline to complete the registration process. But the local Electoral Registration Office could fine you £80 if you don't register when asked.
You may be allowed to register anonymously.
The full register is an authoritative list of those registered and allowed to vote in elections. It is the list that is used to produce the polling cards that are sent out calling you to vote and by the officials at the polling station when they record that you have voted (obviously not how you cast your vote). If you "opt out" of this list then you are ...
Trademark is a type of protection that prevents others from selling products similar to yours with your trademark.
So if a political party had a business selling T-shirts, sure it could trademark its name for the purposes of a clothes brand, and have protection there.
But could it trademark its name to prevent other political parties to ...