How exactly do they work? Is there only a limited amount in the game which means we have to carefully decide who to use them on?
Short answer. Grails are to make your weaker favorite servants viable. Use them whoever character you like. You like Gilles saber? Grail him to 90. It's used to make servants you really want, but are weak in stats stronger or just give general attack buffs to other (Jalter).
Yes it's limited in game and some only have 7 or 10, but it's used like that.
See above. Mechanically, it increases a servants level cap, allowing them to become a little stronger than usual. This will not put them on par with servants of higher rarity though. A 3-star servant grailed all the way to level 90 will still have lower stats than a level 90 5-star servant.
This is something done purely to show 'hey I like this servant'. You get one for each singularity completion and from select events, so yes they are very rare and very limited in number.
