三人行，必有我师(焉)

that one rule above, that 就 needs 了

Hah, and meanwhile I remain extremely confused by 了, so I still don't really know what the deal is there. XD

This might not be helpful at all, but to me it seems like there are only two meanings of 才: "only then" and "just now." To me at least, "only then" already implies that something is "later than expected," so I don't see that as a separate meaning. Basically, I think of "[X] 才 [Y]" as "only when [X] then (finally) [Y]." So for instance, you have these listed under separate rules, but they can be translated in almost the same way:

他晚上十二点才睡觉
I think of this as "Only at midnight did he (finally) get to sleep."

老师来了，教室里才开始安静。
I think of this as "Only when the teacher arrived did the class become calm."

A couple more examples from the question set:

在外面工作了， [才] 知道在家好。
Only after you've worked outside will you know it's good to be home.

他很少回国，两三年 [才] 回去一次。
Only once every two or three years does he go back to China.

And I think for the other meaning, "just now," 才 usually does really translate pretty straightforwardly to "just" (or "just now," or "just then" if the context is in the past).

他才上课。
He just (now) went to class.

他来这个地方的时候，[才] 十九岁。
When he came to this place, he was just nineteen.

Bob [才] 来上海半年
Bob just came to Shanghai half a year ago.

Though I have to say I still find it a little confusing that Mandarin uses a single word for both "only then" and "just now," since I'm used to these being separate things!