Wash me, but don’t make me wet!

I would use a mixture of three different idioms in the examples you've given. Precisely, I'd translate them like this:

So, Frau X is a member of the Green party and campaigned for the closure of nuclear power stations. But last week she joined a demonstration to protest about the siting of a wind turbine in the field where she walks her dog. She wants to have her cake and eat it too.

So, country Y is happy to benefit from EU funding but not prepared to abide by decisions democratically made by the EU parliament. They want the milk without buying the cow.

So, Frau Q is a minister in the left-wing party that campaigns for comprehensive public education, but she has just registered her daughter at a private school. It's a case of "Do as I say, not as I do.".

So, Herr Z is a member of the right-wing party that agitates against migrants, but he has employed a refugee to clean for him. It's a case of "Do as I say, not as I do.".

Note that none of these translations is structurally quite the same as the example you showed, in which "Wash me but don’t make me wet!" was just used as a standalone sentence after the description of the behavior being commented on. I don't think we have any idiom in English that has the same meaning and is used in that manner.

Some notes on the differences in meaning between the three idioms used:

Have his cake and eat it (too)

tends to be used in reference to cases where somebody wants to achieve a combination of outcomes that is logically or practically impossible, but doesn't recognize this. It doesn't imply any hypocrisy, and can be used in situations where there is no moral component at all. Using it in reference to your Green politician, then, suggests that she is too stupid or too deluded to recognize that power must be generated somehow and that you cannot coherently oppose all different means of power generation simultaneously. (If that wasn't the characterization of her you were going for, and instead you wanted to suggest that she perfectly understood that reduced nuclear power would require more usage of other power sources, but that she wanted to make sure that other people suffered from the consequences instead of her because she is selfish, then this idiom wouldn't really apply - you might want to instead accuse her of being a NIMBY).

The cow and milk idiom in the second example is most usually seen in the form

Why buy the cow if the milk is free?

and, in that form, is typically used as advice to women not to let a man have sex with them too easily since this will eliminate the incentive for him to commit to a relationship. The key idea here is that the "milk" represents some nice-to-have benefit and "paying for the cow" represents what a responsible person would do in exchange for that benefit. The precise phrase "milk without buying the cow" is less common, but still used - see https://www.google.co.uk/search?q=%22milk+without+buying+the+cow%22.

In the case of country Y, then, the idiom implies that they are being irresponsible - that they are taking a benefit while refusing to pay the price that morally ought to come associated with it.

Finally, the idiom

Do as I say, not as I do

is used to characterize people who profess a moral ideal but then don't live by it. Your third and fourth examples fit this well (although neither of the first two do).

nimby (noun) ... originally acronym for 'not in my back yard'.

(plural nimbies) (disapproving, humorous)

a person who claims to be in favour of a new development or project, but objects if it is too near their home and will disturb them in some way
The nimbies were out in force against proposals for a high-speed rail link.

• Oxford Advanced Learner's Dictionary 9th edition © Oxford University Press, 2015

The popular expression as the essence of hypocrisy

Do as I say, not as I do.

first appeared in Table Talk, a 1659 posthumous work of John Seldon, criticizing the clergy for preaching moral behavior which they themselves do not observe:

Preachers say, "Do as I say, not as I do."

It lacks the paradox of the German expression, but makes up for it in brevity and strict parallel structure.