What is the meaning of のか in the following sentences? Does it have the same meaning as のですか?
漢字はどう正しく書くのか、どう正しく読むのか、彼らは時々迷います。
参加するのか、参加しないのか、ここではっきり返事しなさい。
Also, I have found sentences of the same structure but instead of のか, they contain only か. For example:
「白」という漢字はどんな時に「はく」と読むか、どんな時に「しろ」と読むか、首をかしげます。
彼はどの大学に入るかはまだ決めていません。
If I change the か in the above two sentences to のか, will it change the meaning?
"What is the meaning of のか in the following sentences? Does it have the same meaning as のですか?"
It appears that you may be confusing the 「か」 and 「のか」 used at the end of wh-question clauses with the 「か」 and 「のか」as question-sentence endings. In all of your example sentences, the 「か」 or 「のか」 is used as the former kind.
「どう/だれ/いつ/どこ/なに/なぜ + Mini Sentence + か/のか」 makes the wh-question clause complete. "how to ~~", "who ~~", "when ~~", etc.
The 「か」 or 「のか」 by itself does not have such an important meaning but it MUST be used in pair with a question word in a question clause. You simply have no choice but to use it.
Regarding the difference between 「か」 and 「のか」 in these types of sentences, it seems that at least in informal speech, we have been using the two more interchangeably by the decade. That is my honest personal observation as a Japanese-speaker. Strictly speaking, though, you would need to use 「のか」 with a wh-word.
If I were a grammarian, I would probably explain the difference by saying that you would be placing more emphasis on the wh-question part by using 「のか」 than by using 「か」. By using 「のか」, you would be requesting a clearer or more precise answer from the other person (or from yourself, depending on the context).
That would be the difference that you would probably be required to know if you were asked about it on a test. If, however, you were just trying to make sense of a 「か」 or 「のか」 used in a wh-question clause in an email or something that is informal from a Japanese-speaker, you might end up reading too much into it.
漢字はどう正しく書くのか、どう正しく読むのか、彼らは時々迷います。
See Section 2, いったい～なのか. It is used to show perplexity
参加するのか、参加しないのか、ここではっきり返事しなさい。
See Section 2, いったい～なのか. It is used to show impatience
「白」という漢字はどんな時に「はく」と読むか、どんな時に「しろ」と読むか、首をかしげます。
彼はどの大学に入るかはまだ決めていません。
See Section 6.
I think のか used in questions (excluding rhetorical usage) generally fall into 4 categories.
You use this の when asking for confirmation. の functions like a quotation mark, which encloses an existing proposition. This kind of の often appears after つまり.
The ～ part can be your conclusion, speculation, or simply hearsay:
(When you hear the rain) 雨が降ったの？
元気がなさそうですね。頭が痛いのですか
This の(ですか) is frequently used in yes-no questions and alternative questions.
Notes: The non-question version のだ/のです is abundantly discussed in research papers. Another related form のではないか can often used in the same context.
You use this の when you feel puzzled and difficult to figure out the answer.
The ～ part can be a wh-question or alternative question, which are often accompanied by いったい. It may also be a yes-no question or alternative question, which is often accompanied by ほんとうに. It is also common to use it to show your curiosity or impatience.
結局何なのですか
いったい何が問題なの
A special の is trivially used in wh-questions. I think it is similar to いったい～なのか type, but the degree of perplexity is very low --- as far as you have used your brain, you can add の. This means, you can almost always add の.
どこへ行ったの
いつ始まるのですか
誰がやったの
One of the restrictions I have noticed is that you do not use this type of の when the wh-word is immediately before (です)か, such as 何時ですか, だれですか, なんでですか, nor when the wh-word modified a noun that is immediately before (です)か, such as 何時のことですか, 誰の財布ですか, etc. In other words, the wh-word must be adverbial. It follows that when you convert a ～なのですか question to ～のは～ですか form, you do not add an additional の.
Another restriction I found is that you do not use this の when it does not make much sense to use your own mind to figure out the answer, for example
お名前は何とおっしゃいますか
(A teacher asks a student) 88と99を足すといくつになりますか
(When taking the order) 何になさいますか
Another special の similar to なんで～なのか, except that the なんで part replaced by a noun. ～ is a yes-no question. This の is used when the focus of the question is adverbial.
この時計は、ネットで買ったの
今日は、車で来たのですか
Thee function of this の might be similar to つまり～なの（か）, which is used to express your speculation and ask for confirmation, but it is just trivially used and does not have much implication.
Just like なんで～なのか, の is not necessary in some cases.
次は高瀬駅に止まりますか？
Notes: This function of の seems related to ～のではない, and mentioned as 否定のスコープ in research books. But I disagree because の is not necessary for this function.
Although の does not change the meaning of a sentence, there are times you have to use it to sound natural.
As far as I know, you should usually use の in cases 1. つまり～なの（か） and 2. いったい/ほんとうに～なのか.
As for 3. なんで～なのか and 4. ～で～なのか, の is usually obligatory in plain forms (だ体), and less important in polite forms (です・ます体). When it does not make much sense to figure out the answer by yourself, you may want to or have to avoid the use of の.
I think the main purpose of using の in plain forms is to expresses your emotions, like curiosity, intimacy, etc. It seems that の and other sentence-final sentences are more common in informal and plain forms than formal and polite forms. For example, you often have to say ～だよ to sound natural where ～です is sufficient.
Sometimes, questions with の sound friendly, while those without any sentence-final particles may come off as curt and rude.
誰がした
どこへ行く
何をする
(どこへ行きますか may still be impolite because asking someone's intention is generally considered improper.)
Even when の is not necessary, some people consider using の as humbler and politer.
先生、この漢字はどう読むのですか
But sometimes, using の when it is unnecessary may make you sound impatient, pushy, or like interrogating.
The Japanese language itself is changing over time. It seems that in contemporary usage, 何で, なぜ and どうして are always paired with の. But this tendency was less prominent several decades ago.
You can follow the same rules above. But indirect questions can be considered as abstract nouns, so sentence-final particles are not really necessary. の can be omitted.
どこへ行ったの？ would this sound slightly feminine whereas のか？・んだか？・のですか？ etc would be used instead for males?
のか and のですか are two versions of the same thing - both are questions with の - but のですか is more formal due to the inclusion of です. I wouldn't say のか has the same meaning as のですか, but it does have the same meaning if you disregard formality - のですか is the formal version of のか. In your example sentences, the のかs are in embedded questions (eg English 'I don't know if...'), and putting のですか is ungrammatical, since you're not supposed to have formality marking inside embedded questions (or other subordinate clauses).
のか and か are not really the same, though. の here is the so-called 'explanatory の', which generally indicates explanations/reasoning/cause/etc. のか thus asks whether or not this is an explanation for something, though it may not be immediately obvious what the explanation is supposed to be for. か on its own just asks whether or not something is the case. An example:
あいつ、やったか？
is a simple question asking 'did he do it?'.
あいつ、やったのか?
asks instead 'is some effect I'm observing caused by him doing it?'.
か and のか are quite often interchangeable, but ultimately they are different.
のか with "cause" and "effect observation". I'd be curious about any source backing it up.
As far as I know, this kind of の shows personal/emotional involvement more than a grammatical function. In the first sentences, they are wondering how to read or write kanjis in general... and whether they should participate or not to something requiring a reply. The の indicates that this question is preoccupying them.
In the next sentence, a specific example is given. It makes sense to me that one is a bit less personally involved about writing and reading 白 than reading and writing kanjis in general. But if it were the final question in a spelling bee contest (for example ;P), then maybe you would wonder with the のかs.
The last example you gave could have had a の to empathize with the student. Without it, it has more of a factual feeling.
There's 2 different usages of か here (but it's not のか on one side and か／のですか on the other one):
か as a subsentence connector.
For example: どの大学に入るかはまだ決めていません, where か serves to connect hasn't decided clause to the which university he goes to subsentence by expressing it as a question, a doubt or similar (expressed in English by how, why etc).
In your 3rd sentence it expresses options (if in English), but that's still similar to this use case.か as a sentence closer. For example: いつ来るか, where か serves to ask a question.whether). As in 参加するかしないか: whether [you] participate or not.You see it actually adds the same nuance to the clause it follows (specify it as a question or a doubt) in all cases, but in the 1st and 3rd cases it is attached to a sub-clause. And as @Sjiveru already pointed out, grammatically the formality of a sentence is expressed on its main clause and sub-clauses use a neutral stance. That's why you couldn't say どの大学に入りますかはまだ決めていません but you could say いつ来ますか.
Now, you can also replace the か particle in the above cases by のか. For case #2 above, it easy to see it as making the question / doubt significantly stronger (as the 大辞林 puts it it's more 問いただす instead of simply 問う).
So you could translate いつ来るのか by When the hell will [he] come...
For case #1 and #3, I prefer to think of it as sub-clause + の + か, each of them adding the nuance they also add when used separately. So with か you express a doubt and with の you express a stronger personal involvement (as @comeauch explained). In case you're not confident on how to generally use の after clauses, check on it first as のか is merely the same in the context of a question. So with your 2 first examples:
漢字はどう正しい書くか、どう正しい読むか、彼らは時々迷います : express a simple fact
漢字はどう正しい書くのか、どう正しい読むのか、彼らは時々迷います : somewhat implies they're really lost at it for example (for this reason something like `彼らは本当に迷います` feels more like a natural context for using `のか`)
参加するか否{いな}か返事してください : rather polite and distant way to ask to answer
参加するのか、参加しないのか、ここではっきり返事しなさい : strong request to take clear action (implying the speaker is possibly frustrated or angry about it already)
And のか is "another" particle, but it stills behaves the same as か grammatically. So どの大学に入りますのかは全く分かりません or どの大学に入るのですかは全く分かりません would similarly be wrong, and いつ来るのですか completely correct (although in this last example the question would be directed at someone else instead of oneself, as the more formal tone necessarily implies).
