Wa alaikum asalam
1)Buses aren't expensive at all, especially if you're coming from a western country.. 500 syrian liras is about 10 US dollars.. I'm not exactly sure, but bus fares are usually somewhere around that price range.. This is from Aleppo to Damascus, btw, don't know about the rest.
2)From Aleppo to Damascus its about 4/5 hours, from Damascus to Amman it's about 2/3 hours, so from Aleppo to Amman it's 6/7 hours. I hear sometimes you have to spend ages at the Syrian-Jordanian border though, so you should probably factor that in as well.
3)The weather isn't too bad, although of course, it depends on what you're used to. It's dry in summers, quite hot at noon, but towards the end of the day the weather cools down a bit, gets quite nice and breezy.