First of all, your right. You should NEVER use the power button to just shut off the computer. In your case, you would have been better off allowing the computer to sit to see if it would eventually unfreeze to be able to, or another option may have been to open your TASK MANAGER and closing down individual programs one by one to see if that would allow you to find what program or programs were making your computer to lock up to begin with. You can also log off as well as shut down with TASK MANAGER. A HARD shut down like you mention forces software to stop without being allowed to close out properly thus in some cases creating broken links and corrupting the software. It sounds to me like you have a second problem as well.
If you say that you hit the power button and the computer sometimes doesn't turn on, you may have a bad power switch OR a bad power supply. Either way, unless you get the power situation figured out you won't need to worry about your software situation if the computer doesn't boot.
If you have ANY kind of computer experience, you know that static can be a killer to a circuit board. So be careful with this next suggestion.........
UNPLUG YOUR COMPUTER 1st !!!!
Open your computer tower. check to see if the fans, mother board, cooling fans, cooling fins, etc are covered with allot of dust. If so, DO NOT WIPE the dust off!! This may have been causing shorts and overheating. Go to a store and purchase the aerosol can of cleaner designed for electrical and computers. (ask for help if need be) to get the right stuff.
Blow off all the surfaces I mentioned careful not to get the aerosol or dust into your eyes. This can create a serious cloud if you have never done this before, so I suggest you do this on the work bench in the garage or somewhere you won't create static. DO NOT wipe anything with a cloth or even place the tower on any cloth while doing this. (STATIC)
This would be a good time to check the wires to the power switch as well as the power supply while the tower is open. See if they are secure.
After you have cleaned out the dust, wait at least 15 minutes before you reconnect the power supply to assure the aerosol has dried completely. Try to boot up the computer.
IF it boots, you MAY now be OK. If not, you may have a bad power button or power supply.
If it does boot to your desktop but then starts to freeze, then you should start thinking you have a software issue. You may have to do a RESTORE to a previous date to see if that resolves your issues. If that doesn't help, I suggest you do a complete restore of either restoring windows, or a destructive restore to put the system back to OEM.