I haven't noticed any issues on the few saves I have where I've just aged past the toddler stage. there was one (and only one) who I did play through toddler and didn't get his potty training finished. As a kid, he seemed to wait a little longer before autonomously going to the bathroom, but he never wet himself. All other traits were fine. The worst that happens is that you actually have to start down at 1 for the child skills (Mental, Motor, Creative, Social) rather than just shy of 3.
You can max parenting starting at kids, but it's easiest if you start at baby and work your way up. Easiest, but not required. Especially if you have the parents more hands on than I tend to do. I've yet to have a parent max parenting if they start with teens, though, and I play on long life.
The primary thing playing through kids gives you is a head start on some adult skills once you've maxed the child skill, and a head start on Character Values.
You don't need to start as a baby to succeed at University. I ran a testing sim that I'd started as a young adult through there (that, admittedly, I'd played for about 2-3 sim weeks before I got Uni) and she came through with straight A's. (Granted, I focused on getting through her classes, so she didn't have the full Uni experience.)
