How do I install Linux Mint on a second hard drive?

Just install the Mint CD and boot it, then select Install Linux Mint from the desktop. After selecting language and confirmation that you have enough drive space available and an Internet connection you will get to the “Installation type” screen.

Does Linux Mint need a swap partition?

For Mint 19. x installs there is no need to make a swap partition. Equally, you can if you so wish & Mint will use it when required. If you don’t create a swap partition then Mint will create & use a swap file when needed.