According to grammar rules, "where they came from and why they disappeared" is a compound subject and should take a plural verb, so "remain" would appear to be the answer in the case of sentence (1). However, this sounds very odd to me:
1. ?Where they came from and why they disappeared remain an open question.
Why odd? Because "where they came from" and "why they disappeared" are actually two questions, and this seems to contradict the singular noun in the predicate, which refers to one question.
Checking GB, I've only been able to find a plural subject with the predicate "remain an open question" in sentences where the subject is not formed by actual questions and where "open question" means "pending issue," for example:
- The true origins of the modern privilege against self-incrimination thus remain an open question. (Source)
- In Serbia, however, the borders remain an open question... (Source)
If instead of "an open question" we had another singular complement, like "a mystery," the sentence would sound much better:
1a. Where they came from and why they disappeared remain a mystery.
Curiously, since "remain" is a linking verb in these sentences (meaning continue to be) the subject could be extraposed and, in that case, the verb would take the singular form because of the presence of anticipatory "it":
1b. It remains a mystery where they came from and why they disappeared.
In the case of sentence (2), there is no doubt that "remain" will be used, because the subject is plural and semantically coherent with "open questions":
2. Where they came from and why they disappeared remain open questions.