How can we ask to people who moved to new neigherborhood/city/country or who started working a new job at a company or who started going new school if they feel comfort with it/ if they managed to adjust themselves in this new culture. How can we ask that in casual way?
I believe the phrase that you're looking for, Murat, is "How are you settling in?"
Here's a link to the phrase "settle in" at Dictionary.com so you can further explore the topic, if needed.
There are a number of idiomatic phrases for this. I would say:
How are you getting along with your new job/school?
How is your new job/school working out?