For example, I want to transform Teen newspaper deliverers and some of the Teen cashiers into Adults or Young Adults and a bartender into a Teen, while keeping their jobs.

I'm already aware that I should not touch certain NPCs like social worker or repo man.

For teens its best to send them to a UNI and let the game age them up, that is if you don't want them to be still doing there job.
As for them keeping there job if you age them up it may or may not change. (Never done it.)

As for a adult to teen should be okay to do but they may lose there job. (Never done it so really can not tell you.)

Cashiers, bartenders, and other service NPCs are safe to fiddle with, but certain jobs are linked to certain ages and you'd have to work to have an adult cashier or a teen barista, using a mod or editing tool to assign the job, and who knows what problems might be caused by that? I suggest experimenting in a test hood, or if you are interested in doing this due to specific sims, to make a back up of the neighborhood before trying it out.

I've aged a mail carrier to elder, later turned him to a zombie. I've also aged paper carriers to adult, then elders. In all cases, they kept right on doing their jobs. If there was any corruption, I saw no sign of it and kept playing the neighborhood for several more generations until I ran out of sims (thanks to norespawn mods). Hoodchecker wasn't an option for me, but use it if you can, just in case.

I just tried out aging up newspaper deliverers and cashiers from Teen to Adult in an old hood, and they remained functional! Now I have to look for uniforms for other age groups

I saw in Cindy's game (Pleasantsims) who was using the NPC age mod that her cashiers turned into adults and kept right on working. So along with what has been said above they do appear to keep their jobs.