Hover the mouse over the constellation of, for example, the arrow and it's name should appear, then just type those letters using the buttons on the sphere by clicking on them.

I managed to get the arm of the statue moving by lighting arrow, pegasus and pisces. But I don't have an arrow to shoot...

Find the two stick pieces and the bucket with the sticky substance in it, combine these and use both papyrus rolls on the fixed stick to decode them.

EDIT: Second stick piece is near the fountain/mausoleum place next to the bridge leading to the broken wall where you can see those arrows flying about.

You have to make the crocodile turn it. There's a hook and a rope in the library

Did you find the rope and the bar? Look at the pillars in the back of the room, you have to mix a potion to remove the paint from them (you can find the recipe in the library). Then put the bar in the pillars and connect the rope+grapple to it. I hope that gets you on track.

Did you manage to identify the different ingredients? That was the tedious part for me... You have to mix them with different ingredients in the top part of that lab iirc (gum etc), by looking what they react to in the notes you've found. The recipe for the sleeping potion is the one found by the slave guy I'm pretty sure. Then he said you had to double the dose. According to the notes I made when I played it, it was: 2x Melukhkha, 2x Ataniya, , 2x Tjekker and 2x Canaan.
But I agree, the potion mixing was the most annoying part of the game.