Are plain noodles (nothing added) healthy?

no they are a lot of carbs

no, noodles are a simple carbohydrate made from white flour.

Healthy describes the condition of the subject of the sentence. Unless the noodles are sick they must be healthy.

It's becoming understood now that carbohydrates (which turn to sugar in the body, though some more quickly than others) aren't "healthy," but you'll still see a lot of people and docs/etc still recommending carbs (though they might recommend "whole grain" carbs or certain carbs less often than others).

Wheat noodles including those with egg (but also rice noodles) become sugar in the body fairly quickly so aren't all that healthy and are not necessary for life. But if they're eaten along with fat and/or protein, they'll become sugar a little less rapidly which is good. Some people also feel that whole wheat noodles are at least better in that regard.

So part of whether they're "healthy" depends on which health hypothesis you believe, and also the amount and frequency you eat, as well as probably what else you eat at the same time.
"Plain" noodles would be considered healthier to the old guard more than noodles with fats and meats on them. But to the new guard, not eating noodles at all, or eating fewer of them less frequently and along with fat and/or protein would be healthiest.

They are a carbohydrate, a requirement in daily nutrition. But there are better carbs, and better tasting ones as well.

Are plan noodles a nutritional food? Do they promote health?
Not really.