Am I playing the game wrong?

[Ronduwil]

I am a new player (highest char @ 36) and I feel like I'm not getting what I should from the game.

I'm not sure what you feel you should be getting, so I'm not sure how to help you with that.

I gain so much experience that I level way ahead of the MSQ which results in being synced, which in many cases removes the functionality of the spells I've earned.

I agree that the syncing removing functionality is super annoying, but it is what it is. As for leveling way ahead of the MSQ, consider focusing exclusively on the MSQ with your main job and then switching to an alternate job for any FATEs and/or side quests that you want to do. I did that for the latest expansion, and I was able to keep my MSQ and XP gain synced for the most part, even hitting a small wall at one point where I actually had to side quest a couple of hours with my main job to progress.

It also deters exploring the world and doing other quests as I detest having spells available then having them turned off. That is such bad form. How can I prevent this?

Level an alternate job alongside your main job.

The game is too easy, near zero risk of death. I believe I've completed 4 or 5 different dungeons at this point and they've been super easy. Open world game play is even easier thus far. Is there a hard mode so a decent challenge can be achieved?

Difficulty progression is a thing in this game, alongside leveling progression. I guarantee many wipes in your future, but I doubt you'll see any below level 47. As far as open world challenges go, you won't find any while questing. There are some special FATE bosses and elite marks that provide good challenges, however. You're just too low level to have encountered those.

I have transitioned from rogue to ninja and lancer to dragoon thus far. In both cases the game play didn't change. I thought the new jobs would be more transformative, did I miss a quest (or quest chain) that would have made it be more game changing?

Once you unlock the mudras for ninja, you'll find the gameplay very different. Sadly, one weakness of this game is that most jobs don't really feel complete until level 70 or so. The paladin rotation, for example, felt super wonky to me until level 80, at which point it finally made sense. I guarantee that playing your job at level cap will feel worlds apart from playing it at low level.