You can remove a lot of preposition and things will still make sense, but that doesn't mean we should. We say that something is/isn't to my taste, and this allows a degree of flexibility as it doesn't necessarily mean a total rejection of it as tasteless, but that we have some problems with it.
So This type of art isn't to everyone's taste means that we can't say that this type of art is a total tasteless to everyone, but can say that it has some problems which cause almost everyone to nearly reject it as tastelessness. Am I right?
Not everyone- if it's not to my taste, I accept that you and others may like it. If it's tasteless, I don't see why anyone would like it.
I don't like Elton John's music, but I respect a lot of his ideas and opinions, so I would say he's not to my taste. I can't stand Madonna and don't think she has any redeeming qualities- I wouldn't say she's not to my taste because it's not strong enough to express my feelings. Others may think she's great, but I can't see where they're coming from.
"Tasteless" is still very subjective. It's not really possible for anyone to say that something is tasteless - you're guaranteed to find someone who likes it, no matter what it is. "Tasteless" is a statement of opinion.
Going back to the original question though, in my opinion, "to" isn't entirely necessary.
Not really, no. "Here's to..." is like a shortened/different form of "Let's raise a glass to ..." If you wish someone or something well by toasting them with a drink, you literally raise your glass towards them/in their direction.
"Here's [a glass raised] to Bob."
Of course, you can't literally raise a glass in the direction of "new beginnings" but the theory is the same.