Learn English – Using the word “whether” instead of “which”

The Corpus of Contemporary American English has 2151 cites for "the way in which" and the British National Corpus has 2574. For "the ways in which", the figures are 2127 and 788. To put that into perspective, both corpora combined barely have 200 cites for "black car".

This goes to show that "the way(s) in which" is not only perfectly grammatical and idiomatic, but also extremely common. So the rule from your book is complete and utter nonsense.

Of course there will always be particular situations in which the phrase is not idiomatic, or even ungrammatical, but then again the same applies to "black car". That is something that has to be decided on a case-by-case basis. As several commenters have pointed out already, the first two of your three example sentences indeed do not work. But the third one actually does. The difference being that in the former two, "the way" is equivalent to "how", while in the last one, "the two ways" refers to two actual mechanisms.

The phrases

I wonder whether he was also right in telling me that I should always arrive on time.

and

I wonder whether he wasn't also right in telling me that I should always arrive on time.

have in common the implicit idea that "he" was right about something else that came up earlier in the narrator's discussion. That's the effect of including also in each sentence. So the question becomes, does the sense of the sentence change at all if you use "wasn't" in place of "was"?

From a purely objective standpoint, I think, the answer is no: Both wordings describe the same thought process and lead to the same possibility that "he" may be right again.

On a subtle level, however, presenting the narrator's wondering thought with the verb "wasn't" implies (to me) that the narrator may once have thought that "he" was wrong about punctuality, but that the previous instance in which "he" was proved correct has led the narrator to reassess that earlier opinion. Using the negatively formed idiom "I wonder whether he wasn't also right" somehow imparts a sense of second-guessing to the narrator's thought that the straightforward "I wonder whether he was also right" does not. At any rate I detect a difference in how the two sentences are pitched, even though their objective meaning is essentially identical.