With reference to the PC being on when you think it is off, you can blame many things for that. First you need to check scheduler to see if some package has scheduled events without your knowledge or permission. Then you need to examine the properties of every package you have installed to turn off the automatic updating that they all like you to have on by default. This can include java, adobe, microsoft update, anti-spyware, anti-virus, dot-net, network time synchronisation and so on ad infinitum.
If you are confident you will be able to remember to update yourself, then turn auto-updating off completely. In the case of Microsoft update, it is best to take the option which alerts you to update availability so you can authorise it when you wish.
Next you can look at the power-saving modes to see if your PC can be woken by various events. A notorious one is wake-on-lan. But wake-on keyboard is a good one that puzzles people who keep pet cats.
If you have power-saving which switches off the HDD after a period of inactivity, that can cause blue screens if the hardware timings are slightly faulty. Your PC can wake up, use the network and try to save something before the HDD has time to spin up. I do not have any power-saving timeouts whatever, I just use my PC or put it in standby.
Next we come to the events causing errors. Here it is my personal opinion that wireless connection of anything is prone to problems, and HP device drivers and software are worse than most. You should check if there is a more recent update of the HP drivers. Also there is usually no need to have any form of desktop monitoring part to the software controlling a printer. It is normally working or not working, and the more things running all the time (especially wireless networking), the more potential for trouble. Just install the driver alone.
At 4 years old, and not obviously gaming when it happened, I do not necessarily suspect overheating as the cause. On the other hand, by all means open the case and with a vacuum cleaner and a non-metallic attachment, do a thorough spring-clean especially around the motherboard and fans for power supply and processor. It can be surprising if you have a smoker in the house...!
Finally your question about turning PC off when not using it, ask yourself why would you want to leave it on? If it is the hopelessly long time it takes to start up from cold, why not do as I do, which is have an icon on screen called 'Hibernate', and has the target
%windir%\System32\rundll32.exe powrprof.dll,SetSuspendState Standby
Not the best option if you keep inquisitive cats, however, but otherwise a tap on a key will have you up and fully working in about 5 seconds.