Hmmm - this is a good question, and it's not easy to describe a concrete rule for it...
To start with let's address your 3 examples, which are the opposite of what they probably should be.
"The government has spent a great deal of money attempting to help the region."
A "big deal" is a colloquial phrase referring to something considered very important or noteworthy. Because of this, "big deal" sounds awkward here. What this sentence needs is an adjective to describe "the relative amount of money." A "large amount" of money is another alternative.
"I don't really care about failing the exam. It's no big deal, really.
This is the inverse situation. Here failing the exam is not seen as important or noteworthy, and is therefore "not a big deal."
You could, however, say "It's no great loss," where "great" describes the relative degree or amount of the loss. "Not a big deal" describes the relative importance of the test.
"It was time for her to make the big decision about whether she should agree to marry him."
Getting married is a BIG decision - it's a big deal. It's important and noteworthy. It needs to be given a great deal (a relatively large degree) of consideration.
I would note that the thing being described doesn't have to be a positive thing - it could be a great loss, a great catastrophe, great suffering, etc. Those all describe the relative degree of the thing. However, a sports team could also suffer "a big loss" if they lose a very important game.