Bishops are appointed by the Vatican, I believe.
As an example, my current bishop, Kenneth Angell, of the Diocese of Burlington, turns 75 in August of this year. Bishops are required to submit their resignations to the Vatican at some time during the year before they turn 75. (Currently there is one bishop I’m aware of who was supposed to send in his resignation in March, and they’re unsure as to whether or not he actually did, as there has been no new bishop announced). The Pope accepted Bishop Angell’s resignation which will occur in August, and he assigned us a new bishop elect, Monsignor Salvatore Matano, who will be consecrated a bishop later this month, become the auxillary or co-bishop from then until Bishop Angell’s retirement in August, and then will be the new bishop.
Bishops CAN serve in just one diocese until their retirement (and since bishops have been made in their early 40s, that could be some 30 years), but they often serve in more than one. Bishop Angell was stationed in Providence, RI and also in Rome before he came to Burlington in 1992, so he’s been here 13 years. Bishop-elect Matano is also from Providence and has also served in Rome.
Hope this answers some of your questions. Bishop Angell will be known as former Bishop of Burlington–he doesn’t get bumped back down to “Father”.