Learn English – Necessity of “that” as a conjunction

"In order to" is a subordinating conjunction. (I.e. a clause that follows "in order to" becomes a subordinate clause, which needs a main clause to make a complete sentence.)

"In order to" generally introduces a "final clause," which is a clause that states a purpose.

The subordinate clause introduced by "in order to" is an adverbial clause, but the phrase itself is not categorized as an adverb.

A discussion of "however," a conjunction that can be an adverb, may be of interest to you. If so, take a look at this blog post.

References:

"In Order to" in English Grammar Today

"Final Clause" in Fowler's Dictionary of Modern English Usage

The standard usage is:

whether this or that

where this and that are both phrases/clauses that should be parallel.

The second clause is commonly elliptical if it's the negation of the first. So:

whether he likes it or he does not like it.

becomes

whether he likes it or not.

Of the examples you gave, "Whether to go outside, or if he should stay inside" isn't considered standard because it is not parallel. It should be "whether to go outside or stay inside". Also, "Whether or not to go outside" isn't standard, it should be "whether to go outside or not".