What is "Where to go" in the sentence "Where to go is the question." Is it a adverbial phrase or a relative clause?
And what is "Why go" in the sentence "Why go when you can stay?" - is it a clause?
Where to go is the question.
I would change to normal word order: The question is where to go.
And I would say the subject is "the question" and "where to go" the complement to the verb form "is" (linking verb to be +complement).
"where to go" has of course the sense of "where one/we should go".
The question may arise how such a shortened question with a question word and a to-infinitive came into being. I would say from formulas such as
1 where it is possible for us to go
2 where it is advisable (for us) to go
3 where we are expected to go.
Such formulas as part of a sentence were so frequently used that they were shortened in the course of time.
As to the question how to call the part "where to go" (Relative clause? Adverbial clause?), it is of course a question clause (shortened) embedded in a sentence and functioning as complement to the linking verb.
It can't be a relative clause, as a relative clause is a sub-element to a noun/noun group as in "The artist who created "I carceri" was Piranesi.
And it can't be an adverbial clause as such clauses answer the question when was it as in
When winter came life got hard for the animals of Manor Farm.
There are a lot of adverbial clauses answering when/why/to what purpose etc.
Re the second question
"Why go when you can stay?"
I would parse "why go" as an independent interrogative clause, and would understand it in one of two ways, depending upon whether it was a direct address (you) or a general question about people and the choices available to them that happened to use "you" to mean "one".
Why (do you opt to) go, when you can stay?
Why (do you opt to) leave now, when you can stay the night here with us? We could drive you to the airport in the morning.
Why walk, when you can ride?
Why (would anyone opt to) walk, when one could ride?
Why take the local, when you could take the express?
"Where to go is the question."
The obvious answer is that "where to go" is an embedded question. If other answerers said this, sorry, I missed it.
Where to go is a wh-clause. It is also the subject of the sentence.
Let us look at the subjetive phrase in parts.
"Where", an adverb of place like 'here', ' there', 'near', 'above' etc.
"To go", a simple infinitive verb modified by 'where'; and infinitives are, amongst others, used as subject of a verb.
Together the subject phrase is adverbial in construction. It explains the 1st.question.
The 2nd. question.
It is a complex interrogative sentence and the interrogation is embedded in the clause, " Why go".
We can view this clause as elliptical form of--Why do you go?--or may regard it in the original form as in the question with the "?" Query mark as its integral part.
In speech, day to day parlance or literal writing, INTONATION matters. It has the power to make a simple statement question or exclamation.
So, " Why go?/why (you) go?/why do you go?-- is the principal clause (interrogative ) of this complex sentence.
In a comment, John Lawler wrote:
Where to go is a peculiar species of compressed wh-clause with infinitive and deleted arguments instead of having a subject and a tensed verb. It almost always indicates a modal in the semantics -- Where to go means where one should go, and Where to go? means where should one go?, for instance. The same is true for relative infinitive clauses -- the man to see means the man one should see. In Where to go is the question, it's a noun complement clause because it's the subject. In Why go when you can stay? it's the main clause, deleting 'should you', though I have no idea how.