Should I capitalize the word "don't" in the following sentence?
Listen... don’t panic.
No, you do not capitalize the word following the ellipsis. I have only ever seen and used spaces in ellipses in academic writing when omitting text from the middle of a quote. For instance, from the Wikipedia article for linguistic ellipses, in the sentence:
In linguistics, ellipsis (from the Greek: ἔλλειψις, élleipsis, "omission") or elliptical construction refers to the omission, from a clause, of one or more words that are nevertheless understood in the context of the remaining elements.
I can use ellipses to remove information and inform the reader of the missing text, as you can see if you compare my new sentence to the original:
In linguistics, ellipsis . . . refers to the omission . . . of one or more words that are nevertheless understood in the context of the remaining elements.
In your case, it sounds like you're writing some form of dialogue. When using an ellipsis in dialogue, you are indicating a pause in the sentence. You do not use spaces or capitalization surrounding the ellipsis. In the sentence you provided, the correct style is: