So I found examples of both 'would you' and 'do you' before the phrase 'happen to'.
Do you happen to have...
Would you happen to know...
Are they interchangeable? Please explain the differences.
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There exists a strange convention in English of less direct constructions (often using the conditional) being considered more polite. One common example is the phrase would like:
Would you like anything?
I would like the hamburger with french fries.
Compare this with the more direct want:
Do you want anything?
I want the hamburger with french fries.
While acceptable in some other languages, this would be considered informal or even impolite in English.
Let's look at your examples:
Do you happen to have…
This construction comes across as slightly pushy. The word "happen" helps (again, notice that this makes the sentence less direct and hence more polite), so here's the sentence without "happen":
Do you have…
This is really what you're trying to ask. And yet, strangely, in English we add "padding" to our sentences to make them more polite. That's how we go from the most basic construction possible from this:
Where is the document?
Would you happen to know where the document might be?