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you should state in written form the bad condition and that you did rent the flat under the condition that the remodeling would be terminated. State that the bathroom is unusable with the exposed wires and leaking water from another room.
Give him 2 weeks to fix the problems (mention an actual date) and state that you will terminate the contract without prior notice if the problems are not fixed within the stated period. Alternatively you will terminate the contract by the next possible date (just in case the landlord or a court finds a reason why you should not be entitled to terminate at once).
Take fotos of the bathroom in order to be able to prove the condition of the bathroom if the landlord later tries to obtain additional rent from you.
No, in Germany a contract can also be done orally. This is different form US law. There is no parole evidence rule.
For the landlord it will be relatively easy to prove that a contract exists -> Would you really have moved in without a contract just to "occupy" the flat?
However, you may have problems to prove that he agreed to finx the problems by June.
Thus, to be on the safe side you should give him a very short period in written form to solve the problem and then move out if he does not manage.
Sorry, I am a German lawyer. You may have misunderstood things on the site.
Of course you can move out and than argue about the content of a possible contract.
However, you asked whether you could move out without problems. As a lawyer I am obliged by law to give you the SAFEST solution. You can decide whether you will want to run a higher risk.