I'll be posting some more I know later.

Some other example questions (You) could answer:
Canon (Lore):
- Character that has appeared the most in a manga (probably Reimu, but need a count)
- Character that has appeared the least in a manga/game (separately if you want). Bonus points if its made relative to the amount of time the character has been shown since she's been introduced, as some newhu would be the most obvious answer here.
- The character that has canonically lost the most
- The character that has canonically won the most (probably Reimu?)
- Character with most relations to different characters (most likely Okina)
- Character duo that has fought each other the most on-screen

Canon Game:
- The most broken, non-bugged shottype in any game (counting bugged is probably MarisaB in MoF?)
- Longest Touhou game if everything is timed out, on Lunatic (Phantasmagoria games obviously won't count) (can separate mainline and decimal games)
- Touhou singular boss appearance with the most spellcards on Lunatic (basically only counting one appearance. Not adding up, say, Yuyuko's Spellcards on PCB + TD)
- The spellcard that spews the most amount of bullets in the series, if timed out. Yes, it's nigh impossible to answer, but it is an interesting question. My best guess is Fitful Nightmare, from Remilia in ISB.
- Longest Touhou Timeout Spellcard (this one I'm almost certain is Profound Danmaku Barrier ~Phantasm, Foam, and Shadow~, from PCB via Yukari, but need confirmation)

Doujinshi:
- Circle/Artist that has made the most doujins. Separate r18 and sfw works if you want.
- Character that is part of the most amount of doujins (likely impossible to properly know as most doujins don't get scanned)
- Character that is part of the most 18+ doujins (same problem with above)
- Oldest known doujinshi
- Once (or still) 2hu doujinshi Author that has had success the most in the manga industry (probably Bkub?)