Very interesting question (for a change)!
This is what I can tell you:
"Manges-tu la pâte ?" refer to 2 things, I believe: either the raw dough (before baking) or the baked dough when fruit or other ingredients (pizza) have been put on top.
"La pâte" is used in singular in that case. Depending on the recipe, you can have: "pâte à pain" for bread and pizza, "pâte sablée" (short cake), brisée" (short crust pastry) or "feuilletée" (flaky pastry) for other preparations like pies or tarts.
Less common, would be preparations called "pâte" because of its soft or crumbly consistence, like "pâte d'amandes", "pâte de fruits" or "pâte d'olives" (... paste).
"Le pâté" is something else, based on proteins, like "le pâté de foie de canard", "le pâté de campagne" (mainly pork) or le "pâté de saumon", etc.
"Danish pastries" are "les viennoiseries" in French (why? I don't know why it's moved from Denmark to Austria...). In this family, you have "le croissant", "le pain au chocolat", "le pain au lait", "le pain aux raisins", "le chausson aux pommes", "la baguette viennoise"... all found in "boulangeries" and "les brioches" as well.
Italian pasta is "LES pâteS" - ie feminine and always in plural.
Hope it helps.
Given that helpful explanation, the English "dough" seems odd. In English dough is raw - the word isn't really used for cooked food (like cooked pizza). So the English translation "Do you eat pastry" makes sense. It would be peculiar to ask someone if they eat "dough" (NB: "the" is not needed in the English translation).