Hello pcpunk - A long time ago, I ran Puppy live from a USB stick and remember having the same problem you are having, setting up a network connection is difficult, but not impossible. I went around and around with it until I finally got it working, I just wish I could remember how.
Your CPU is not too small or too slow, my brother's HP Mini laptop has a 1.66 GHz Intel Atom processor and it runs Linux Mint 17 just fine. It's not really fast starting up or loading certain applications, but it doesn't crash. The big difference between his system and yours is memory, his has 2GB and yours has 1GB. This becomes even more of a problem for you when you try to run a "live" Linux session from a CD or USB stick. When you boot a live session, the Linux kernel, the desktop, and other system processes are loaded directly into system memory. Then when you open other applications, like Firefox, even more memory is used to load that. And while Firefox is running, it is caching files to system memory as well. Eventually, your 1GB of memory gets used up and you crash the system. If you had Mint installed on your hard drive, it would make a difference, especially where the browser's cache files are concerned, because they would be cached to your hard drive instead of system memory.
Which brings me to something I was thinking about Tuesday night after posting here. There are two things you could do when running a live session that would really help your system as far as memory usage. The first is, right after you open Firefox, go up to "Edit" and select "Preferences". Then click "Advanced" on the new menu that appears. Then select "Network", check the box for "Override automatic cache management", and set the cache size limit to zero. This will prevent the browser cache files from being written to system memory and hopefully save you from more crashes when running Firefox.
The second thing I thought of is this, you should create a Linux swap partition on your hard drive. When you boot into a live session, the swap partition will be detected and mounted during boot. This would enable the system to swap files from memory to disk while running a live session. Since you are running Windows XP, you are not able to use Windows Disk Management to shrink your C: partition and do this. One utility that can do this for you is MiniTool Partition Wizard. You can download it here: http://www.partitionwizard.com/download.html
Be aware of one thing: this utility, if used carelessly, will wreck your existing Windows system. You could end up using your backup image from Macrium Reflect if you are not careful. Please read the instructions on the website before using this software.
First, you want to shrink Windows C: partition by 1GB(1024MB), please read these instructions: http://www.partitionwizard.com/help/res ... ition.html
Note: You want to create unallocated space at the end of C: partition (the right side), not the beginning! (the left side!)
Second: You want to create a new partition in this 1GB of unallocated space and format it as Linux swap, please read here: http://www.partitionwizard.com/help/cre ... ition.html
If you are having problems creating bootable USB sticks with your Linux .iso files, try these two utilities for Windows:http://www.linuxliveusb.com/http://rufus.akeo.ie/
And to install more than one .iso to a USB stick, try Yumi: http://www.pendrivelinux.com/yumi-multi ... b-creator/
If everyone is thinking alike, then no one is actually thinking.