"They have a big Italian restaurant."
Translation:Nekik van egy nagy olasz éttermük.
"Nekik" can be omitted here, it will be understood from "éttermÜK".
"Van egy nagy olasz éttermük."
Very interesting! From étterem - éttermük? Or does it come from another name for restaurant? Maybe the em ending is turned in this case, instead of prolonging the word? (instead of étteremjük?) How is for the others? éttermem, éttermed, ?
"Étterem" + "ük", the last vowel is dropped. It happens with a few words. "Terem", "verem", "szerelem", etc.
And yes, you guessed right for the others. Not ALL of the suffixes though!
Is the indefinite article "egy" necessary here — i.e. would "Nekik van nagy olasz éttermük." also be acceptable?
Yes, that sentence is possible, though the word order is not neutral. In speech you would need to put emphasis either on the word nekik, in which case your sentence could be translated back to English as "they are the ones who have a big Italian restaurant", or on the word van, and then it could be translated as "they do have a big Italian restaurant".