A:AnswerI will answer the first part of your questions. It does require a wifi network and it does require and internet connection. You lost me on your question regarding phoning home everytime.
A:AnswerYou are able to connect two devices to the HDHomerun. So if you have a laptop and watching tv on it, you can connect a TV or another device to it. Or watch TV on two TVs at once.
A:AnswerThere is a separate version for cable. The one here will only record over the air channels. I believe it is called the HDHomerun Prime that will work with Cable, but it only works with non-encrypted channels so check with your provider beforehand.
A:AnswerThe HD Homerun connects to your router. You would not need to connect the antenna to your TV unless you also want to be able to tune channels using the TV's tuner.
A:AnswerYou can watch TV shows on your Apple computers. You can also watch on your phone or tablets, but can't watch it directly on a TV without extra hardware and software. (e.g. Plex)
A:AnswerThis device is NOT for cable setups. It is for over the air (antenna) setups. If you want to use a CABLE CARD with COX you should look into the HD Homerun Prime. You should also check with COX to see if they support cable card setups using the Prime.