"To go"

No.

Although we have seen an older Italian woman (older than we are, that is) who was also eccentric in other ways but well known to the restaurant, ask to take half of her meal home. The restaurant seemed to be ready for her request.

I don't think people order every course except maybe on special occasions. We typically order only antipasti and primi.

We got a side eye when the meat and cheese board was enough and we asked for the pizza to go. Then again the waiter was a bit chilly from the beginning

I live in Italy and I take leftovers home all the time. Nearly every meal I eat out, because yes, the portions are large. The restaurants have always had containers or at least aluminum foil, and I have never gotten any static or funny looks. And I didn't pretend it was "for my dog."

Customs may have changed since Covid, making take out more common, but no one will think twice about your request.

As noted, customs may have changed. When, in 1987, I presented a paper coffee cup ("We Are Happy To Serve You", blue Parthenon) to Italian morning-bar people (Reggio nell Emiglia), and politely asked for "Da portare Via, per piacere", they looked at me like I had two heads. Why would I want to give up the good fellowship of their cafe stand-up dining patrons? Not to mention a hot, ceramic cup?

Pretty normal nowadays. I know our eyes are sometimes bigger than our stomach, but I often skip antipasti in Italy (it’s not something Italians eat at home on a daily basis, only for special occasions). Antipasti have the tendency to fill you up before you even start the real meal. if you order only a “primo” (first course) or a “secondo e contorno” (second/main course with a side dish), you may be able to finish the meal. If there is more than one of you dining, you can also split the dishes. That is also normal and, more often than not, they split the portion into two half portions for you and bring them in two separate dishes. The awful practice of charging extra for splitting dishes, as I've seen in several restaurants here in California, is not at all common in Italy, and actually I’ve never seen it done there.