Which sounds better?
There is water and butter in my fridge.
There are water and butter in my fridge.
I think it should be: is.
But what if we said:
How much flour and butter is needed to make a pizza?
How much flour and butter are needed to make a pizza?
In that case, I think the plural verb: are, is the correct choice, which means (I think) there is a contradiction between both sentences.
Your question involves many different topics. It seems that the main gist of your question is about subject-verb agreement. But your two sets of examples are quite different from each other.
For instance, your first set of examples are declarative clauses that involve the existential construction:
Which sounds better?
There is water and butter in my fridge.
There are water and butter in my fridge.
I think it should be: is.
The grammatical subject is the dummy pronoun "there". And so, to figure out the number of the verb, er, well, there are a lot of bogus "rules" out there, taught in schools and in grammar usage manuals and style guides, and so . . . well, if you are in school or at work, then you do what you are told. But if you writing for yourself, then you'll probably have to rely on your ear--except you might be an EFL speaker, and so, er, . . .
For more info on the existential, there is this post:
Your second set of examples are interrogative clauses that do not involve the existential construction:
But what if we said:
How much flour and butter is needed to make a pizza?
How much flour and butter are needed to make a pizza?
In that case, I think the plural verb: are, is the correct choice, which means (I think) there is a contradiction between both sentences.
You can try to force some sort of "rule" for this type of sentence--and I've seen a bunch of bad "rules" being passed around the internet by pedants--but if you're in school (or at work) then you'll probably want to do what the teacher wants. Otherwise, the context and the writer's intent is most likely what's going to determine what's most appropriate.
In any case, the two sets of sentences have different types of constructions, and the coordinated noun phrase (a coordination of noun phrases) is not part of the subject in the first set but is part of the subject in the second set--and so, there doesn't really seem to be any contradiction if different verbs are preferred in one set or the other.
The number of a noun phrase is in itself a rather complex topic. And that number can be overridden, grammatically or notionally. Context is usually king. Let me correct myself: Context is king.
You can also say "There are both water and butter in the fridge".
It depends on the way that you parse the sentence:
?how much (flour and butter) are needed to make a pizza
how much flour [is needed] and how much butter is needed to make a pizza
I can see both as valid parses, but the first seems a bit stilted to me. There's no contradiction, it's simply a matter of how you read the phrase. English allows for this type of ambiguity.
Saying "there is butter and water in the fridge" and "how much flour and butter is needed" sounds more natural and logical to my ear.
Consider that what is implied in those phrases is the amount.
And so:
"There is water and butter in the fridge" means just about the same as
"There is an indefinite amount/supply of water and butter in the fridge."
Likewise:
"How much flour and butter is needed to make a pizza" amounts to saying
"What amount of flour and butter is needed to make a pizza."
