R27: Theaters have an intendant or a general manager who's the administrator of the whole shebang (although he might be answerable to a board, as at the Met).
Then the theater may have a conductor who is the music director. The MD is closely involved in planning seasons and has a lot of say in the singers on the house's roster and in the casting of specific shows. He will personally conduct a lot of the big-ticket items, new productions and important revivals.
What the MD doesn't conduct (because he's not interested in the repertoire, or it's not his specialty, or just because there's always much more work in a season than any one person can handle) is overseen by other conductors. They might be prestigious guest conductors in their own right, or they might just be up-and-comers or veteran journeymen, so their influence will vary. When the Met had Riccardo Muti as a guest conductor for the first time about 12 years ago, the general manager let him choose EVERYTHING: the opera (he picked one the Met had never performed), the whole cast, the stage director, the design team, even whether it got an HD movie broadcast with cameras (no).
The theatrical side of opera is the province of the stage director, but stage directors don't typically get a choice in which singers are in a production. Many of them would LIKE to, and they would probably prioritize good acting and looking the part over the singing, but they tend to be stuck with the theatrical abilities of the singers whom the conductor, casting department, and administration want to feature.
The most powerful conductors have a lot of influence over the whole show. Levine at the Met, for example, had very conservative tastes, and so at the peak of his power in the '80s, '90s, and early '00s, there were a lot of new productions by his established favorite directors.
I hope that helps a little. It's a hard subject to get into, because there are a lot of variables and exceptions. A young conductor coming to the Met to conduct five routine Bohèmes won't have influence over the casting and won't get a great deal of rehearsal time. A famous conductor coming to the Met for a new production will be a different situation.