Unfortunately, the issues you are experiencing are symptomatic of an internal hardware defect, often a faulty power supply or main/control board, though there are many possible failure points. This issue could be a manufacturing defect, but is also often caused by a power event (shock/surge, prolonged over/under voltage condition, etc).
This could also be the result of age related wear and tear; as a 2007 model tv, that type of failure wouldn’t be surprising.
There is one step you can take here, both as a possible fix, as well as a diagnostic evaluation:
1) Unplug the power cord from the outlet
2) Wait at least 1 hour (60+ Minutes)
3) Press and hold the power button (on the set, not the remote) for 60 seconds
4) Keep holding power button while plugging back in, and for another 60 seconds after. When plugging back in, go directly to the wall outlet, bypassing any power strips or surge protectors.
While unplugged, it would also be a good idea to thoroughly clean the tv by vacuuming off the grille/vents and blowing it out with compressed air to remove any dust that may be contributing to overheating.
If this reset process does not work, it would unfortunately confirm an internal hardware failure.
This is something that would require repairs to correct; if you're thinking about a do it yourself repair, you'll need to be comfortable and skilled with electronics repairs, which would require dismantling the set (removing the back panel of the tv), diagnostics with a digital multimeter to test voltages inside, changing out boards as needed, and possibly desoldering/removing and soldering/replacing individual components.
If you’re considering repair at a local shop, you’ll find a typical repair to be around $400 for diagnostics, parts, and labor. These pioneer plasmas were some of the most complex tvs ever made; they are nearly impossible to find parts for and many techs won't touch them or just aren't qualified to do so.
For a tv of this age, size, and resolution, replacement will likely make more sense. Even though this was a very high end and expensive tv when new, you could replace it with a brand new LED tv with a fresh warranty and twice the picture quality for as little as the $500 range these days.
Depending how you’d like to proceed here, I can help with finding parts, finding a repair shop, or recommending replacement options. Just let me know.
Best regards, ***** ***** for the news,