Definition: Given the Question and Answer pair, detect whether the answer is acceptable or not. Denote acceptable answer as "yes", otherwise denote as "no". An acceptable answer is consistent with the context in question, detailed, long, and uses general knowledge of English. Answers that are not definite and suggest a variety of options are not acceptable answers.
Input: Question: Should I write it like this? Or perhaps like this? Should I write it like this? or perhaps like this? What about after an exclamation mark or semicolon? 
Answer: The word after a question mark (or exclamation point) is capitalized if it starts a new sentence. If the next word does not start a new sentence, because the question mark is part of a quotation, a title, an interruption, etc., then it is not capitalized. Usually such things are set off with quotes, dashes, parentheses, or italics; but not always. Your option #2 is a single interrogative sentence with two parts and a question mark at the end of each part. This is rare today, but it was apparently a lot more common historically, when sentences were longer. It is not at all hard to find examples with the word or: Do you not know what beauty is? or do you not know whether you think him handsome? (1814) Who lives in a more elegant style? or receives more illustrious visitors? (1815) Should he anticipate the savages? or should he lie still, and abide their assault? (1822) Could you have done as much? you or any other man? or any other woman that ever breathed? (1833) Does the science of medicine prevent death? or the institution of government prevent crime? (1847) "Where are we scheduled to walk? or are we to have something to say about it ourselves?" (1896) These examples are all from the Corpus of Historical American English, which only goes back to 1810. There are no doubt plenty of examples in later decades I stopped looking but this way of writing questions has all but disappeared. I rather like it, but to some people it's just wrong, and to others it might seem literary or pretentious. Update: Consecutive sentences in the prologue of Shakespeare's Henry V contain a non-final question mark and a non-final exclamation point. [...] But pardon, and gentles all, The flat unraised spirits that have dared On this unworthy scaffold to bring forth So great an object: can this cockpit hold The vasty fields of France? or may we cram Within this wooden O the very casques That did affright the air at Agincourt? O, pardon! since a crooked figure may Attest in little place a million; And let us, ciphers to this great accompt, On your imaginary forces work.

no