Just to be sure I’d check if he really means it.

Look at it this way. You’re a guest in someone’s house. It’s nice you contribute to utilities, but that doesn’t make you the owner or part of the family. And when there is conflict in the family, you’re the first one expected to leave the house.

Have you considered sea and then land transport? It will take a lot of time, but you will be able to take care of your dog personally.

Rent in Italy, work in Switzerland

It’s understandable. In such a case Hungarian citizenship won’t help you much at the recruitment stage. It will play its role when signing a contract, but on a CV it unnecessarily complicates your situation.

Your situation seems a bit (unnecessarily) complicated. If I understand correctly, you have dual US/Hungarian citizenship and currently live in the US and you are looking for a job in Western European countries.

You should know that most IT jobs have been outsorced to Central and Eastern Europe. Finding a job in Germany, for example, is much more difficult than in Hungary, Czech Republic or Poland. Especially when you’re not German.

If you like Hungary, I can advise you to look for a job there. The earnings will be nominally lower than in Western Europe or the USA, but compared to the cost of living you will be satisfied. In IT you should earn 2x or 3x the national average, which gives you a very comfortable standard of living. You will be able to rent an apartment, everyday expenses will not be a problem and you will be able to save. Of course, when you come back to the US, everything will seem expensive.