How to categorize this phrase. Relative clause, Interrogative clause, Adverbial 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".

YOU

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.

ONE

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.