Depending on what country your office is located + personal preference it could be that every project you do is in another country.
Regarding your questions:
1. You do not move office. McKinsey has global staffing with a global PnL. You stay employed with your home office. There are short-term transfers to different offices but this has nothing to do with project experience abroad.
2. Taxes depend on the country. E.g., if a European is staffed in another European country then there are no implications. If a European is staffed in the US, part of the tax needs to be paid in the US, etc. Nothing to worry about in general.
3. Generally yes, however, there are nuances to it. If the country is close, you can fly home every week (the standard operating model), if the country is further away you need to find a rhythm that works for you, e.g. flying home every 3 weeks for 4-5 days or so. Alternatively, in the latter option, you can also fly in your spouse for a couple of days to spend with you, rather than flying home. Often, if the whole team is far away from home, they spend the weekends together and travel within the area.