The mystery woman's quest is finished in the remake, Diana's story is left purposely without closure. I'll just spoil it for you, Diana is Albert's older sister. To understand her backstory, you need to play his chapter, but the game is set up in a way where the two can pretty much never meet. If you keep her in your party, you'll never meet Albert and likewise, she can't be recruited if Albert is the chosen MC or in your party. Diana appeared in Albert's ending, but it's unknown if they ever meet again. Made slightly dickish considering Albert and Diana can both do missions for the one character who knows they are both alive. From what I've read elsewhere, Diana is a better character than Albert stat wise.

The only two quests I'm aware of that were not completed in the original but finished in the remake involve the mysterious bar lady, and the necromancer. They both also dealt with the two missing Fatestones as it was impossible to get all of the stones without hacking the game since neither of these missions were completed.I wouldn't be surprised if there was more cut content. From what I've read elsewhere online, there are a lot of unused assets in the code and heavy implication a few other surprising characters were meant to be recruited which entails more missions that were likely cut in the original. The remake adds back a bunch of stuff and actually has a larger roster of recruitable characters than the original which is kind of daunting to say the least.