What do you mean by "pings"? I've used LG and Nokia phones on a Verizon subscription plan for the last umptyump years and have not experienced something I'd call a "ping".

A couple of thoughts:
- if your phone has a GPS built in, you can set the GPS for 911 only (rather than on all the time). This will mess up some phone features (like VZ Navigator, which depends on the GPS), but might "leave you alone" (that's how my phones are set).
- if you're talking about a minute-by-minute beep, many of my phones have a feature that makes the phone beep shortly before each minute is up so you can end the call before you incur another fractional-minute charge. I don't know if this is the problem, either, but it can be an annoying "nanny".
- if you have one of the dwindling number of handsets that can use an analog signal, some phones will beep or otherwise notify you that you are moving from digital coverage to analog (or back to digital). You can either shut off the notification or, usually, force the phone to digital-only if that will meet your coverage needs.

As for the phone getting hot, some phones do. I had a Samsung years ago that got almost too hot to hold after a 30-minute conversation. My Nokia and my current LG (Voyager) have been much better in this regard. My LG VX-4400 has been kind of middling.

Good luck.

Assuming you are talking about the tracking/locating feature, there are supposed to be privacy regs that control how that info is used. On the other hand, there are any number of government and quasi-govt agencies that probably have the ability to do so also. Not too much you can do.

Prepaid is the only way to go if you want real anonymity, and even then I'm sure there's stuff they can do sometimes. You would have to buy them for cash etc etc for anonymity, that's why criminals and terrorists et al like them.