1. Is taking a lower salary just the cost of working at a startup as compared with a FAANG?Yes.
2. Is it stupid to decrease your salary by that much when changing jobs, or is this something that everyone does in exchange for working at a smaller company?
No, it only happens when you downgrade from FAANG (or from a small set of unicorns with similar compensation ranges). Whether it is "stupid" or not is up to you.
3. If I know they can't pay me as much as I currently make, but I still want to work at a startup, what should I tell them when they ask me what salary I'm expecting? My current salary? Something lower?
See the advice in sibling comments (e.g. romahn's). Whatever you do you'll have to be prepared eat a 30-50 percent salary decrease. What's up to you is how you spin this fact in your negotiations.
Getting back to what you said at the top:
The money is great, but the job is boring.
By all indications, this precisely what FAANG pays you for - to stick around and put up with whatever they throw at you.
It was to test your appetite for this level of boredom that they made you do all those LC problems, after all.