Level 1. In the very beginning, where you are supposed to get on the roof? He doesn't grab it. Is there somewhere else I'm supposed to go? Or is it a bug?

I think the jump you are referring to is in level 2. (the radio tower) Try setting your game to "always run". It should make the jumps easier. And then run and jump. When you want to walk up to the edge of a ledge without falling off, hold the shift key down in order to walk, not run.
You also might want to set your game to an easier setting. I think the default setting is harder.
Also, install the patch.

Dragon, where exactly ARE you in Level 1? More details please. As for the radio tower which is in level 3 Tian Shan River, make the jump to the roof by standing as close as you can to the right of the tree on the left and then making a running jump. Do pick off the guards below first though.