the proof of the pudding is in the eating
Question 1: What is the role of the?
Question 2: Is it possible to replace the with "a"?
(I am assuming that "proof" means "test".)
The principle of this idiom is that each individual pudding can be judged on its taste. Some might be good, some might be bad - you'll know when you eat it.
Without the definite article, 'pudding' would be a mass noun and mean all pudding everywhere. That wouldn't make sense in the context.
Yes, I suppose you could replace it with the indefinite article and it would still make grammatical sense. It might even make more sense in modern English. But this is an idiom - an old saying - and as such people say it the way it has always been said.