According to Wiktionary these words were borrowed by Hungarians from Slavic languages. Hungarians liked to borrow feminime forms of Slavic adjectives. I guess it occured because of Slavic wifes of early Hungarian warriors.
No, I don't think so. That is my first thought. But on a second thought, it might.
Just as the English "dear" can mean "close to the heart" or "a heavy price", the Hungarian "drága" also means both. "Drága" is the common word for "expensive", "precious" and "dear (to the heart)". It may very well have... no, it actually did come from a Slavic word "drag". And Russian is also a Slavic language.
Az means the or that right? It said "that restaurant" was incorrect, but isn't it a correct translation too?
No. "that restaurant" would be az az étterem.
az "that" and ez "this" are used together with a(z) "the".
In English, we don't say "that the restaurant" or "this the book", but in Hungarian, we do.