Yes, it´s very common for African Nations to fire their coaches in the middle of the season. First I was invited for Ivory Coast, after their sucessfull World Cup Qualification, and later I was invited to Cameroon, a game before the end of the African Cup Qualification.
I´ve also noticed that some European managers have been fired in the middle of qualification. Poland, Denmark and Israel did this. I guess you have to look frequently at the job vacancies if you are interested in national squads.
Yes. I was managing TSV 1860 Munchen. If a manager is fired, or quits, the national squad will appear in the Job Security window as vacant. There you can apply for the job, and keep the team as well, if they accept you. For Ivory Coast, I applied. For Cameroon, I was invited. Both can happen.