Is despatch a misspelling of dispatch that made its way into the dictionary? Could I use dispatch instead in the above example? "Dispatch" is the most common, and therefore most preferred, spelling, but both spellings appear side by side in the revered Oxford English Dictionary (see the work cited entry below). "Despatch" isn't a misspelling, just a less preferred spelling.
Do you pronounce both of them the same? These words are really two spellings of the same word; therefore, the two spellings share one pronunciation.
Are they a homophone of one another? The two spellings have the same pronunciations, but may not qualify as homophones because they are, in fact, the same word spelled two ways.
"dispatch | despatch, n.". OED Online. December 2012. Oxford University Press. 5 March 2013.