>Surely Japanese characters should not be treated the same as Mandarin ones

No. In Japanese, how you read/pronounce a character depends on context. Sometimes they are the same as Chinese, sometimes not.

Take mountain (山) for example.

Using the Chinese pronouncation it is "san". 富士山 (Mount Fuji) is ふじさん "Fuji san"

Using Japanese pronouncation it is "yama". 山登り (Mountain Climbing) is やまのぼり "yamanoboru"

(and don't call me Shirley)

I can't speak Japanese, only some Chinese, but I'm wondering if whether to use the (Chinese) Onyomi or (Japanese) Kunyomi pronunciation in Japanese is related in any way to whether the 山 comes first or last in the compound. If it comes last as in 富士山 "Fuji san", the grammar matches the Chinese, and so does the pronunciation ("Fushishan"). If it comes first as in 山登り "yamanoboru", the grammar is opposite to the Chinese (which would also have the 山 last, i.e. 跑山).

PS: Isn't り pronounced "ri" and る pronounced "ru"?

As a rule of thumb, I have learned that Onyomi is usually when a kanji is part of a compound word and Kunyomi is usually when the kanji is by itself.

Yes, I typoed that and it's too late to fix it. り is "ri", not "ru".