Here's what I've tried with no success:
- Unetbootin: It keeps hanging.
- Linux Live USB: I only get an empty folder as the result
- Universal installer: it hasn't worked.
Can I use my Ubunu CD-ROM in persistent mode?
You cannot do this with a LiveCD
CDs, for the most part, are designed to be read-only once they've been written to. Even rewritable CDs typically require CD writing software, because most other applications will see the disk as a standard CD and assume it's read-only. This means that you cannot save your printer setup to the LiveCD.
You can install Ubuntu to a USB thumb drive
It's actually deceptively easy - take a standard live CD and boot to it with your USB drive plugged in, and install Ubuntu to the USB drive like you would to any other hard drive (be sure that everything points to the USB drive). You can then boot to it by selecting the USB drive in the boot device selection menu when your computer boots up. This method does format the drive to Ext4, so Windows can no longer read it, but it makes the entire drive available to the Ubuntu install, just as though it were on a regular hard drive.
Your printer driver issues aren't all related to your LiveCD issues
A 500 error is an error coming from the server that Ubuntu is trying to download from. It means that something is wrong with the server itself. There's nothing you can do about that error from that download mirror for that particular driver, except wait a while (anywhere from a few minutes to a couple of days) and see if it comes back. However, the LaserJet 1000 series is old enough that you should have a generic driver option available, which should allow you to install it. If that option isn't available, or doesn't work, then simply download it directly from the HPLIP website (it also includes instructions on installing it manually).
Saving the settings, of course, will require a writable medium, which I've addressed above.
You need to create a Live USB for persistence as live CD cannot store your settings and data.
And for this you can simply use Ubuntu's own Startup Disk Creator or MultiSystem.