I just came across 峠 and noticed that it has no on-yomi. Wow, does that mean the kanji does not originate from China but is Japanese-made? Are there many kanji without on-yomi? So far, I only noticed some kanji don't have kun-yomi.

Yes, as far as I know, 峠 is what is called 国字.
You can see a list of them here http://kanjitisiki.com/kokuzi/

Funnily I made a thread about that topic around a week ago. It has some interesting numbers.

http://forum.koohii.com/showthread.php?tid=12494

tehfriendlyghost Wrote:So far, I only noticed some kanji don't have kun-yomi.

Every kanji has a kun-yomi (in theory), but many of them are not on the Joyo Kanji list.　Some of those non-Joyo kun-yomi will pop up from time to time in modern writing, others you will never see unless you study kanbun.