"How do you set changes permanently to linux with a dvd, or can't you?"
Assuming you used a live CD to play with Linux, then no, you cannot. Or you have to include the adjustments you made into the Linux image file before you burn it. If the changes were minor, you could create a script that makes these changes and run that script after you boot from the installation media.
"Do you still need an antivirus with just linux mint on dvd, or only if you run an accessory HD?"
No OS is completely safe. Viruses do exist on the Linux platform. But with common sense, you won't catch one that quickly. And if you booted from CD and do not have links to the hard disk inside the system you are using to play with Linux, there won't be a virus after you reboot this system.
"Can a root kit still invade the mobo if you only run linux mint on dvd?"
Yes, if the rootkit is designed to do this, then it doesn't matter if you boot from any type of installation media. It will store itself in some hidden nook from the BIOS/UEFI and because of that, it doesn't matter anymore how you boot your computer.
"I do have a 500 GB SATA HD I could spare for this.
Would that enable me to save changes to the OS?"
Preferably disconnect your main hard disk before connecting the spare hard disk. then do a proper install on the spare hard disk. When that is finished, you can make changes to your heart's content. These will be stored on your spare hard disk. With only the spare hard disk connected, you can be sure that you won't mess up the partitions of your main hard disk. After Linux Mint is installed you can hook up the main hard disk again an Linux Mint will recognize the main disk and enable you to access the partitions on the main disk.
"Would I need antivirus protection?"
If you don't surf with common sense, open everything you see and run as root the whole time...then by all means, get anti-virus software. But if you don't, then changes of getting a virus are small as there aren't many Linux viruses around to begin with. And by default Linux is setup to not run as root. The surfing with common sense...that is the weakest part on any operating system. Linux won't protect a stupid user from him-/herself.