The meat is usually fried along with the onions until it acquires a bit of color, but it’s not usually browned in the manner you might see French kitchens brown something. That said, I have seen some people brown their meat for goulash. It’s usually veal (which is typically made into a paprikash, which is essentially a pörkölt with the additional of sour cream and flour) that I see the suggestion not to brown it at all.
Yes. Celery stalks are not typically used in Hungarian cooking, at least as far as I’ve seen my five plus years living there. It was generally pretty difficult to find celery stalks, although that situation has changed in the last decade.