Your iPod can only be synced with one computer at a time. If you try to sync it with a new computer/iTunes library, it will replace the contents of the iPod with whatever is in the new library. You should probably disable the autosync functionality first by going to Edit -> Preferences, clicking the Devices tab, and enabling the prevent iPods,iPhones, and iPads from automatically syncing option.
Before doing anything else, authorize the new computer with your iTunes Account. In iTunes, choose Store -> Authorize This Computer and enter in the correct credentials. Either copy a backup file from your old computer to your new one or create a new backup of your iPod in iTunes before letting it sync.
Then right->click on your iPod Touch from under the Devices section in the left hand pane of iTunes and choose Backup. You might also want to take a look at this article to see what it all included in the backup.
Now onto synced content such as music, videos, photos, etc. For iTunes purchases you can copy them back into iTunes by choosing File -> Transfer Purchases. For all other nonpurchased iTunes content such as music, videos, and photos synced to the device, see this older post from another forum member Zevoneer covering different software available to assist you with this task.
Once the backup has been made and all other synced content such as music, videos, and photos are back in your iTunes library, restore your iPod from that backup you made earlier. Here is more on backing up and restoring your iPod.
If you do happen to lose any purchased content, you can always redownload it from the iTunes Store again at no cost via iCloud. See this article for information.
Yes. Based on discussions here no one has been able to find a free program for synced photos.