1. You select a plane, use "rebase" and click on the carrier. Note that you have to be in range just like when you're rebasing to another city.
2. That depends on how much more science you're making. The focus basically just tries to get as much science from the city as possible. That means that it'll prioritize working tiles that generate science, like tea resources or Gilgamesh' Ziggurats, and it'll prioritize specialists in the campus more than it would otherwise. It probably won't matter much, though, and if you're making, say, 300 science a turn, then making 10 more science might well be less than a turn off of research. You might be a turn faster over the combined time of 3 or 4 techs though. My advice. however, is that focusing on science, culture or faith is nearly never worth it, and if it is, you're better off locking tiles or specialists manually.
3. If that is the difference I'd advice scaling down a difficulty level. The AI starts with bonus units depending on difficulty level, and from Emperor on that also means a second settler, which allows them to start getting everything up and running faster. On top of that, they earn 24% more science. If you're playing a decent, Emperor level game, you should be close to the AI's from about Industrial or Modern Era on, maybe one that is ahead of you by more than an era. It should be noted, however, that the AI tends to "beeline", which means that they research very deep without researching cheaper techs that come earlier, so while they have a tech or two in the information era, they may still be missing a bunch in the modern era.