Changes in port numbers, routing tables, connecting...

My very simple network: DSL modem/router & iPad 2,2. Router SSID off, login changed, WPA convoluted encryption.
Began to have network login (and other) problems at least a year ago. I have no IT/network background and hope someone here has a diagnosis. At this time I am not connected to icloud, have no push enabled, have nothing running on iTunes. The Network Analyzer app shows Port 62078 is always listening, data streaming, ports 49153-49165 established at power up, bz.push active, etc. I've looked up some of the ports and see soft phone (which is double scary because my cell has rec'd odd calls/commands) and Xsan. All of my apps have crashed and now have .ips at the end. Each time I look up .ips file, I find a different explanation.
Most recently I was unable to login to my bank account and after reaching the bank and changing PW found an http: file in my address book that would not open because it was a "local" file. Huh?
I'd like to know how to establish and maintain a closed, secure, private network without having the creepy feeling that my iPad screen is showing up on somebody's Apple TV.
Can anyone point me in that direction? Please.
I do have a MacMini which I disconnected because file sharing and wifi was turning itself on.