If I reset my HP pc back to its' facotry settings, will the OS still be installed on the computer?

It will (more or less) return to the state you'd purchased it in, which will include the OS.

The OS should be installed already after settings your to factory settings but if it doesn't boot from your Os disk by pressing f12 as soon as you turn on the computer.This would delete everything you have saved on your computer and you may need to install some drivers too.This would get your computer back to what it was like when it was first turned on.

Yes

If you are restoring the setting that actually means formatting your hard drive and reinstalling your OS along with your drivers and other bundled software. So please take your backup before hitting that restore button

Yes it will still have the OS. It is when you format it gets rid of your OS.

Yes.

press F 11 at start up>> follow instructions

it will be restored back to how it was when you first purchased it, so yes the OS will still be there

yes as long as its the OS you have when you first purchased it.

if you are referring to a system recovery - where you put a CD in to restore to factory settings, then yes it will.