I'd say הציפור שייכת לאברהם or הציפור היא של אברהם are both ok in daily conversation. I think pretty much the same. הציפור של אברהם can also be ok. In reading it's ambiguous, but when you say it in conversation, with tone and context, it would be understood. I'm almost tempted to take back what I said about it being archaic. Maybe I overthought it now... :-) maybe someone else would like to comment.
In the previous lesson I had to translate Is the food yours? and I tried ?האוכל הוא שלך and then ?האוכל שייך לך which were both rejected. The only accepted answer was ?האוכל שלך, which I thought was ambiguous (at least in reading) for the same reason you mentioned in your comment above. Are my translations fine? If so, which of the three questions would a native speaker be most likely to ask?
I agree with almog, all forms sound okay to me. למי זה שייך? = של מי זה?
the only problem with your הציפור של אברהם is that there is no predicate. it might not be a mistake, but if someone said it to me, I would expect there to be one... its just like "avraham's bird". like, ok, but what about it? :p
but you can say, for example הציפור הזאת של אברהם or זאת הציפור של אברהם. once you give more information (its THIS bird youre talking about), it makes your sentence more clear.
It's a funny thing why European languages say "Abraham" and "Jakob" although they do have the consonant /v/. I believe it's because they inherited spelling and pronunciation of Biblical names from classic Latin (or was it Greek?) translation of the Bible, which collpased some Hebrew consonants because classic Romans (Greeks?) didn't have them. Same with ש of שמעון, שלמה and many other names, which became "s".