Since I also moved here to Hungary, I can related to most of the points in this story!
Especially the language issue, I definitely felt it at first. Everyone in my surroundings speaks English, but I agree with you, we feel the pressure that we are basically forcing them to switch to English when they could, otherwise, just speak HU. I ended up hanging out more with people from work. Even though most of them are Hungarian, since it's an international company, everyone has a great level of English.