If you’re asking this question at all, I’d say you’re probably not ready for a minimal installation without a desktop environment but it’s totally possible.
I’d recommend just going with a normal distro and installing a desktop environment from the get-go as that will be the easiest way to get Linux set up.
That is neglect. Please call whatever authorities are relavent in your area (list of DHS numbers for the US: https://www.childwelfare.gov/state-child-abuse-and-neglect-reporting-numbers/) and they can help you.
I hope you and your sister find the support you need.
Welcome to the club! I hope you find Mint to your liking!