I have a 7-year-old AT computer with a 233mhz processor, and it runs Firefox browser just fine.
Memtest86 is what I always use to test RAM. It runs without starting the Operating System, so it doesn't matter which operating system you have.Memtest Link
Unfortunately, this requires the ability to download the file, and then burn an image to a disk. It's a very simple program - you simply set the BIOS to boot the machine from the CD and it starts and runs as long as you want. (Press the Escape key when finished.) Best to run it for at least several passes or overnight.
I'm wondering what sort of condition your power supply is in, and if it's sufficient for the things you're running. Intermittent problems are often caused by a bad power supply. Bad memory tends to cause the computer to freeze.