No, it's definite for "stad". It's technically "staden", but the "de" gets swallowed in casual speech so people sometimes write it to match in casual writing. I don't live in the area so I don't know about the place's history, but judging by a quick glance at Wikipedia the name started as a nickname for the area, so that might explain why it's informal.
That's a special form that's used only for males (like a man, a boy, or a male dog), but it's entirely optional. It might be a bit more common when used as a vocative (as in "Kom hit, gamle man", "Come here, old man"), but that's mostly me guessing. All in all people probably won't raise an eyebrow if you use "gamla man" instead.
Farfar is your paternal grandfather, whereas, if I do not confuse it after I actually should know better, morfar is your maternal grandfather. And if you switch the two parts of those terminologies, you will have respectively your paternal and your maternal grandmother. It's just two morphemes put together like blocks of a wall.