This phone is an android phone with hardware buttons on it. At the bottom end of the phone, you'll see 3 buttons. One that looks like a back arrow, one that looks like a Home, and one that looks like a stack of windows. If you press the Home button and it doesn't do anything, press and hold on the power button. You should see a screen that shows up which has "power off" as an option. Tap on that, and let the phone shut down. Then press and hold the power button again to start it up again, which should hopefully get you to your home screen.
I've used plenty of apps on my phone while it was charging for well over a year now and I've had no issues. The only thing which will be different is that the phone will charge more slowly, because you are doing things on the phone which require power too. If you want to quickly charge your phone, it is best to completely shut it down and then plug it into the charger. However, if you want to use the phone while it's plugged in, there will be no issues.
If I remember correctly, you can press and hold Power + Volume Down to access the bootloader screen on these phones with the phone powered off. Try that, and if it does get to a crude-looking screen with a bunch of options, use the volume keys to select Recovery and then press Power to select it.
It could be a variety of issues. If it's a blue screen saying that, that means it's an issue with Windows crashing and being unable to start. If you press F8 repeatedly while the Compaq logo is on the screen, you should get options to start in "Safe Mode." Start there, and see if the computer will boot in Safe Mode.
Now, some of these D620s shipped with NVIDIA graphics chipsets. If you remember back in the day, there was a lawsuit and recall regarding these chipsets failing prematurely due to heat-related damage. One of the symptoms of a failed graphics chipset is the screen turns on but stays blank. If you plug in an external monitor using the VGA port and you get garbled or no output, that's a guaranteed failed graphics chipset.