Will skill levels transfer over whenever sims are aged up? (help needed)

You can find a sim's job level by going to the Careers panel. Hover your cursor over the job icon at the top left corner. The tooltip will tell you the sim's level.

The best thing to do, when you aren't sure whether to do something, is save the game first and then try it. If you don't like the results, you can just go back to the Main Menu without saving.