There aren't really any areas of the city to avoid during the day, but the Bigg Market (just off Grainger Street...it's just an area of pubs and clubs) can get quite rowdy at night...especially Thurs - Sat. I wouldn't say it's particularly dangerous to be there, just more a hassle of fighting through crowds of drunken people...not really the best place to go if you're looking for a quiet drink.
I think the Laing opens at 10 and closes at 4:30 or 5, I think. It's on the corner of John Dobson Street behind the city library.
Yep, Gateshead is just on the other side of the river. Newcastle is on the northern side of the Tyne and Gateshead is on the south.
The pronunciation of Geordie is just as you described.
Also, just thought of The Gate -- an indoor entertainment venue -- there's a movie theatre there and several restaurants and pubs. We spend a lot of time there! Like everything else mentioned, it's in the city centre -- if you're on Grainger Street, there'll be a Boots (chemist) on the corner -- just walk down that street and you'll see a big building with "The Gate" in big yellow letters.
And don't forget about the markets -- the Grainger market can be accessed on several places from Grainger Street and the Green Market is in Eldon Square shopping centre. If you head toward the Gate and walk down the street with Cafe Nero on the corner, you'll find yourself between the two markets -- I find that the easiest way to go back and forth between the two.