Not sure what the real difference is, but Ad Aware and Spybot often find things the other doesn't, hence the advantage of using both.
I have never had any problems letting Spybot remove stuff. Obviously, if you have programs which rely on Adware / Spyware to run, they won't work if Spybot does it's stuff, but then you're probably better off looking for alternatives anyway.
Not sure about ZA. It should warn you if a program is attempting to access the internet, but some spyware are just tracking cookies, so ZA won't tell you about everything.
Hope that helps, but a proper techie would probably advise you better.
