Hey,
I can provide you with information regarding the dual boot, my HDD is dual booted with Ubuntu and Kali.
Firstly, you must have a live version of your OS (Mint) on a USB flash drive or CD, it must be the same version as your installed version. Go ahead and boot your live OS and enter the live trial mode.
Open up GParted and click on your main HDD (usually /dev/sda1), then right click and choose resize/move. resize the HDD into a partition that is your desired size, this may take a long time to complete depending on how big the new partition will be... i split my 1tb hardrive in half and it took around six hours.
As for the LUKS, I am sorry but i can not help you there as i have no previous experience with it. There may be a way to encrypt your chosen partition from within the installed Mint OS. I am sure it can be done in Ubuntu if i remember correctly, so it is most likely possible in Mint too.
