Leaving a postdoc after 3 months (for another)
If you believe you can handle the bureaucracy quickly, so as to get paid, do that. Depending on the local culture (Europe is not very homogenous), this might involve calling people and going past the official channels, or just doing everything you are required to do and waiting.
After having secured the payment, or immediately if you think it would take too much time to get the payment, talk to your supervisor about the situation. Your goal is to exit the situation so that your current PI understands and accepts it, so that they will be available for recommending you later. If this turns out to be too difficult, then your goal is to exit with as little damage to your reputation as possible.


If you judge your PI to be a reasonable person, just tell them that you are having difficulties with another postdoc, and try to give a fair and reasonable explanation. Also tell that you do not feel at home in the city. If they are a good person, they will understand, you can try to find the best possible solution in an unfortunate situation. The group dynamics in their research group are their problem and they can do something about it. The nature of the city cannot be helped.


If you have not tried talking to the other postdoc, you might also try that, if you have the courage. Something like "Thank you for helping, but I can do this by myself; could you, for example, do [other part of the project] so we would get results faster?". Or simply telling them that their behaviour is bothering you; check, for example, "nonviolent communication" with your favourite search engine, or maybe parenting books, for relevant advice. Tell factually what happens and how it makes you feel, and ask for concrete actions or how the other person thinks the interaction could be improved. Do not blame. Stick to facts, as much as possible. Do not attribute intention, attitude or sexism to the other party, as this will not help your cause, even if it is true.