I can't replicate any of those bugs with v4.0.2. You appear to be playing with the German translation and also have a lot of other mods installed. I'd suggest switching to the English version (the one you can download here) to make sure you're on the most recent mod version and then trying to apply to the career with only Aliens Reloaded and Core Library to make sure its not a mod compatibility issue.
You may need to go to a Discord support server like Deaderpool or Sims After Dark where they can take a look at your full mod list and make suggestions.
Edit: I was eventually able to replicate this issue via game corruption, so you can probably fix this by doing a game repair (or a reinstall if you're on Steam.) If you're not able to do that you can also use cheats to add the trait "Kayaker: trait_isAlienInfiltrator" to your sim and they'll be able to take a mission from the computer. Once they take their first mission they'll be in the career.