Learn English – Using “of” before “whether” and general usage of “whether”

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".

As per Wikipedia on algebraic conjugates, it would be correct to say, "A is the conjugate of B," as you have noted. And of course, B is the conjugate of A, given that conjugates come in pairs.

Based on my experience, it would also be correct to say, "A is conjugate to B," with the reverse also being true. No mathematician's eyebrows would go up.

On the other hand, I don't know anyone who deals with these issues on a regular basis who would say, "A is conjugate with B." Instead, they would say, "A and B are conjugates," as suggested by @PhilSweet. I admit that this is on the soft side for ELU, but that's just the way it is.