Which P4? my workshop/basement PC has the 541, bit newer but it's still ancient to run W7 on it, Vista wasn't even considered as it's basically a beta OS that was released on a rush.
I run 2K on it as XP crashes the same way as yours, tries to load but restarts, probably a driver conflict of sorts even though the motherboard is XP compatible, I used it to play the test build of Half-Life yesterday but its main use is working with floppy disks.

Only works offline though, no networking.

Mine was made in 2003. The blue screen it gives stays there and I can't press any keys and the HDD stops.

HDD suddenly stopping in fact may be a hardware problem, dry capacitor(s) at power supply outputs causing +5V/+12V rails going out of spec, or having high ripple.

Title says it all however, I want to also mention the fact that this PC I'm asking y'all about has Windows XP Home Edition. Will I be able to upgrade it to Vista or
7? Also, not planning on using the PC as a Linux machine.

As far as i can remember, you can have a DDR-1 version or a PC-133 version.
The first, you can upgrade to a max of 4GB of ram (if supported), the second, i doubt.
So, with IDE bus HDD, low RAM capacity and single core processor (with bad performances), it is a triple NO.
Keep XP.

Also, beware of capacitor plague, which was common 20 years ago.