Using a question mark mid-sentence
I've mainly seen these in older literature, usually used in dialogue, but sometimes in rhetorical essays. It is rare in common and modern writing. I'd advise against it.
You should either capitalize or rephrase/repunctuate:
Would you like the drapes to be white? Or perhaps something off-white?
Would you like the logo to be centered? At the bottom? Left off entirely?
Or:
Would you like the drapes to be white; or perhaps something off-white?
Would you like the logo to be centered or at the bottom? Should it be left off entirely?
The former change (capitalizing) is more informal than the latter suggestion; also, the latter can be legitimately tweaked in several places based on context.
I randomly picked some old books and found examples in every one:
From The Way of the World by William Congreve, 1700:
Right, lady; I am Sir Wilfull Witwoud, so I write myself; no offence to anybody, I hope? and nephew to the Lady Wishfort of this mansion.
From History of Tom Jones, a Foundling by Henry Fielding, 1749:
Nay, I will venture to go farther, it is being in some degree epicures: for what could the greatest epicure wish rather than to eat with many mouths instead of one? which I think may be predicated of any one who knows that the bread of many is owing to his own largesses.
From School for Scandal by Richard Brinsley Sheridan, 1777:
O Fie--Sir Peter--would you have ME join in so mean a Trick? to trepan my Brother too?
From The Three Musketeers by Alexandre Dumas, 1844:
"How? know you again? Did you ever see that man before?"
From Marvels of Pond-life, by Henry J. Slack, 1861:
What are they? animals or vegetables? or something betwixt and between?
It was quite common in the 18th century books, less so in in the 19th century. Perhaps now that it's the 21st century, it's time to die out, but I kind of like it. It feels like a link to the past, an archaic but valid use of the symbol.
Oh mid-sentence question mark! Will you die out? with none to resurrect your use?
I often see this in exercises in elementary mathematics textbooks. First it asks a question, then adds some variants afterward that are only fragments:
Farmer Jones has a horse pen that is 12 meters by 10 meters. What is the area of the pen? the perimeter? the width?
I believe in these cases the question mark is closest in function to a semi-colon. If you know what I mean when I say this, then feel free to use it with the same sensitivity as you do with a semi-colon (and ignore Kurt Vonnegut, he's so hipster it hurts). A sort of pause: "would rather the curtains be red? or some other colour?"
If you don't feel confident in using a semi-colon properly, then perhaps it's best to ignore it and stick to the most general usage, which is like that of a full stop.