I have four cities now, and there's still plenty of high quality land to settle. Should I produce more Settlers?
By all means keep exanding. I'd say at lest two cities more before you start developing the capital.
Finally do some Granaries?
Yep, while your capital is busy producing settlers, other cities can grow and whip granaries. Pasargadae can easily 1-pop whi granary thanks to its good base production, but you should not stagnate Susa attempting to do the same thing there. Chop if you want to build granary in Susa with 1-pop whip, but don't stagnate. Working that gold won't hurt your capital, because it can't grow while building settlers anyway.
Do I need a Library in Persepolis?
Of course you do, but the right order would be: settlers, granary, library. One of the reasons you want your first cities to be productive early on, is that you need other cities than capital capable of producing workers and settlers at decent rate, so you can develop your capital. That's very easy with imerialistic leader, because it makes a settler cost just 10

more than a worker. Any city with a granary and good food can build settlers at the rate of 1 per ~6 turns.
At some point I will also have to hook up strategic resources and build some troops to avoid wars, right?
Wrong. You don't build units just to let them obsolete. That would be a horrible waste of hammers and gold. Only build units when you are preparing for war. Of course, you are never completely safe without getting AI's to pleased/friendly. And that is one of the reasons why early wars are good. You gain something through conquest and by the way make yourself ready for potential DoW's.
I have marked some possible city spots on the map, are these good, or should I expand towards Shaka? I really like the spot
with Ivory, Horses, Stone, Oasis and fresh water lake in the south.
I don't really like that spot, that city would have to stagnate, probably at lvl 4, working oasis, horses, ivory and gold. But it might actually be a good decision to claim all those resources with one city, because there is no food in the whole area around the ivory. You need to scout the coust, though. There can be some seafood, that would change everything.
Your c1 dot is plain horrible as it kills the fish. May be 1W? You're about to finish Alpha and backfill Fishing and Mysticism, a city 2W of the copper could take it from the capital and quickly build a monument and granary. In the mean while the capital could grow on food-high tiles. I would make it my 6th or 7th city.