Contrary to what you seem to think, wouldn't and won't are almost never interchangeable.
The simple negative won't is used for future negative actions or for refusals.
- I won't go to the store tomorrow if it's raining. (Future negative.)
- I won't go to the dance with you. (Refusal.)
The negative wouldn't is used for counterfactual statements, and for future statements embedded in a past-tense narrative.
- I wouldn't shout if I were you. (Counterfactual)
- He said he wouldn't like it. (Future embedded in past narrative.)
In every case here, replacing won't with wouldn't results in something either ungrammatical, or it changes the meaning of the sentence.
Edit: An additional requirement for will/would is tense concord, which means that subordinate verbs in a complex or compound sentence must agree in tense with the main verbs. This applies to the two halves of an if/then construction, as well as to verbs in relative clauses. For this purpose, will is considered to be present tense, and would is past tense. So you see things like:
In this case, the distinction between will/would doesn't carry any semantic weight, but is required by English grammar. Swapping will and would in any of the preceding sentences results in an ungrammatical utterance.
There is absolutely nothing wrong with
Since it's raining, wouldn't it be a good idea to bring an umbrella?
Look at a few other constructions to see how natural it is:
Since you're going to the store anyway, wouldn't you like to pick us up some ice cream?
Since our success is uncertain, wouldn't we be well advised to consider how to minimize our risks in case of a downturn?
The first clause assumes a fact, and the second balances it with a speculative counterpoint. That is grammatical, unexceptionable and, incidentally, one way to make a sentence interesting.
Best Answer
As FumbleFingers notes in a comment above, the expression is part of an online comment responding to a January 25, 2015, news story at BizPac Review titled "Prisoner who escaped ISIS captivity tells NBC: They want something bigger than 9/11." The comment reads in its entirety as follows:
No doubt the commenter means to say, in the second sentence, something like
or
Unfortunately (for the commenter) it's very easy to get turned around when combining forms of negation, so that you end up saying (from a strictly logical point of view) the opposite of what you mean. Here the commenter has already navigated through one "Do you really doubt" sentence unscathed, although that one is a somewhat simpler construction.
The second sentence builds rhetorically on the first—but then it goes off the rails, perhaps because the commenter isn't sufficiently confident that "Do you really doubt" establishes an adequately negative overlay on the relatively complicated expression that follows. Or maybe the commenter simply forgot that the verb earlier in the sentence is doubt and not think. Constructions of the form "Do you really doubt that..." are fairly treacherous, especially if you're the kind of commenter who doesn't read what you type before pressing 'Post comment'.
In any case, do you doubt that the commenter meant wouldn't in the sense of "would" in the second sentence? I doubt it.