For something to spy, an application has to be running. Backups don't store applications so restoring from backup can't put anything of that sort back. It would have to re-download from the App Store and you would have to intentionally re-start it. Frankly I don't know if you could install that kind of thing even if you wanted to. Apple has iOS so heavily sandboxed ( = regulates what things can and cannot do with other parts of the phone) that I don't know if you could have a package doing all you claim without messages popping up all the time about you sharing your location, etc. As Lawrence Finch said, you'd have to be using a jailbroken phone that would let things run hidden and also communicate secretly with other services on the phone.
Now there are many features on an iPhone that can do some of what you say but they are there with that purpose, such as sharing your location and other things with people you have deliberately set up to do that with. My wife can see where I am and I can see where she is because we have location sharing on.
The classic situation where things show up elsewhere is when two people use the same Apple ID.
What exactly showed up on the computer? Were you using the computer previously? Did you have iCloud turned on so it was syncing things with an account on the computer?
Data spikes -- probably the phone was backing up to iCloud when that happened, or maybe you have it set to update automatically.
Running warm -- unless the phone is turned off there can be background processes running and updating. If I leave my phone turned off the battery can last for days, but if I leave it in sleep mode it only lasts a day even if I never use it at all during that time. The phone is keeping a bunch of apps running and updating the data (e.g. fetching my mail) and uses up the battery unless I have the things shut off completely. Even then it will still show slight drain because it uses the clock. My iPads which I use once per month are almost always dead when I next want to use them and they are turned off all the time.
Echos -- I don't know. I remember those from long ago and long distance calls. Hang up and call again.
If you are truly convinced something spooky is going on, set the phone to factory settings and start afresh with setup.
Restore your iPhone, iPad, or iPod to factory settings - "A factory restore erases the information and settings on your iPhone, iPad, or iPod and installs the latest version of iOS or iPod software." - http://support.apple.com/HT201252