If you can post the model of the HP laptop, we can learn more for you.
You should find the model information on a sticker on the bottom of the machine.
It sounds like ESC or f10 will take you to the BIOS configuration utility.
Have you tried f8? You may need to press repeatedly, or press and hold.
This will take you to the option to boot to Safe Mode.
If this is successful, let us know. If not, let us know.
Next... you obviously have internet access from another computer.
There are two utilities you should download and run.
These utilities will check for errors in 1) your memory and 2) your drive.
Since they run outside of windows, trying to run them will give us a good idea if it is a hardware issue, or if it is a problem with your OS. (the f8 question may leave this an open question, but this will help get a definite answer, in addition to trying to isolate hardware faults)
To Test Memory... Using Memtest Please follow this guide...
Please note that (unless you start getting errors reported) you will need to let this run a long time.
A good way of doing this is to let it run over night. In each 'pass" there are a series of eight memory tests that are run, each more complicated than the previous. We used to advise letting it run a minimum of 7 passes, unless you start getting errors.
We now advise at least 10 full passes. If you have errors reported, you may want to re-run the test with only one stick of ram installed, checked each stick until you isolate those that are faulty. That way you can limit your replacement to those that actually need it.
Also, I would like for you to run some hard-drive diagnostics, but I need to know more in order to point you to the right tools to download.
I may or may not be able to find out your drive maker from the model information you supply about your laptop.
Sometimes that information may be found in your Bios (bios sometimes detects it).
I will tell you how to determine your drive maker this way, after you tell me more about your laptop.
If I / we cannot determine the drive maker another way, we may need to have you open your drive bay cover.
Is the machine out of warranty? If not, for you to do this next step would void your warranty.
Are you comfortable opening the machine?
Do you understand the hazards of static electricity and know how to avoid damaging your computer with static?
Depending on your answers, I may have additional steps for you before we have you download your drive diagnostics.