"I'd like to ask you: where are the scrolls in Saiyuki coming from? Are they just released over time in battle?"
Lots of places: regular shops, one-time-purchase-only shops, in treasure chests picked up during battles, hidden in destroyable environmental things like rocks, and the characters come with one or two pre-equipped already.
"Is there a limited amount?"
There are 40 total - 8 of each of the five elements.
"Does one learn abilities over time with the scroll equipped, or do you immediately have all abilities associated with that scroll/class upon equipping?"
Each scroll grants the character one spell the instant it is equipped, and the scrolls don't level up. The player does have to think about which scrolls to equip since each character only has 6 slots for scrolls or accessories. The characters' proficiency with elements level up as you use them instead of the scrolls leveling up.
"Are there miss-able scrolls?"
Yes. I know Clone and Chef's Paradise come to mind. I don't remember any others off the top of my head. Oh, the Emporer guardian is missable too. Some characters are missable: Genshi and Reikan, and it sounds like Kinrei, Ginrei*, Taurus, and Tessen** although I've never had it happen to me in my four or so playthroughs.
"Are some of them a bitch to get?"
No. If you do all the sidequests in Chapter 2, you'll get Clone in Chapter 3. And if you enter every Shop, Blacksmith, Post, Chemist, and Dojo you'll get Chef's Paradise in either Chapter 2 or 3. The key is to re-visit the Shops, Blacksmiths, etc. in Chapter 2 after recruiting Kinrei and Ginrei. For Genshi and Reikan, just open a FAQ when you're prompted with dialogue choices in the cutscenes. The rest of the scrolls aren't missable. And for the Emperor guardian, it's also pretty simple: open every treasure chest in the battles, destroy Taurus's throne and open the chest that appears, and enter every Shop, Blacksmith, etc.