Is it considered rude in America to...?

It should not be. But it will be embarrassing

Yes, it is uncouth to ask if you can finish someone else's plate, even when you are home. You can do this with family or close friends, but not with dinner guests someone is trying to impress. You can offer to help clear the table and once you get in the kitchen you can wolf down some of their leftovers (as long as no one sees you), or scrap the plates and put the leftover food aside to be eaten later (but make sure you wrap it so no one assumes it's just trash. If you are out with people and they don't want to finish their plate, you can ask if they are going to finish and when they say no, you can ask if it'd be all right for you to "doggy bag" it to take home. You can explain you have a dog or you an explain that the food is still good and you could have it later. Some people may be a bit offended, but others would feel that at least their leftovers are not going to waste and as long as you aren't afraid of getting some germs from their pickings, go for it.

Eating someone's leftovers at dinner is not rude but it would get you talked about in a derogatory way.

If you were at your parents' house, was there something that prevented you from having what the guests had?

As for hating to waste food, as long as you were not paying for the food, it's not your concern how it's used or wasted.

You could have said if you get a box to take it home I will finish that for you.

Though I think its kind of gross to eat something that others were eating.

It is rude, do not ask. You just have to contain your excitement for the food and accept it is not yours and it never was.

Why would it get you talked about in a derogatory way if it were not rude?

Yes that would be rude and a little weird too to ask guests if you could finish thier food. Family its okay, but never your parents guests.

it it was a casual lunch or something, sure just ask. if its a fancy dinner, better not. a lot of people like to take their leftovers home anyway.

Ask if you can clear away the dishes and then chow down.

In general I would say yes,

friends and family...go ahead..ask... total strangers...not so much