The challenge you'll find here is that you have 3 different video sources (apple tv, ps3, and dish network) and most of the inexpensive HTiBs are severely limited in the number of inputs available.
The two options you listed, in fact, have NO additional video inputs AT ALL. In this case, you'd have to run the video from each source into your TV (provided it has enough inputs), and run the audio from each source separately into the Sony unit.
It seems that each of these has BARELY enough audio inputs to satisfy your needs - 1 optical digital, 1 coaxial digital, and 1 analog RCA input.
One other thing to keep in mind, is that "wireless" surround speakers is a largely misleading term. Yes, there is no need to run speaker wires from the receiver around the perimeter of your room to the speakers. Instead, the Sony receiver wirelessly transmits the audio signal to a separate receiver unit for the surround speakers (it's about a foot long). This unit (which needs to be positioned near an outlet for power) then sends the audio to the speakers via speaker cable. So there will be speaker wires running to the speakers themselves.
Your budget is a bit limiting, and if the "wireless" approach to surround is an absolute must-have for you, then I'm not sure of any better alternatives.
Without the wireless requirement, there are a number of more feature-rich home theater systems from Onkyo that coud be had. Amazon.com has the
Onkyo HT-S3200
for less than $350, for example.
Jason