Can a Soldier/Service member be non-deployable on account of a family member/dependent?

You can be, but there are time limits involved, if you remain no deployable beyond that time you can be separated.

If you get a compassionate reassignment, you won't deploy while on that assignment. Otherwise, no. That's why you have a Family Care Plan that can ensure they are taken care of when you have to deploy, go to the field, TDY, etc.

I had to got SLC 2 months after my daughter's transplant...used my FCP. I was on a compassionate reassignment a few years ago that luckily the unit had a slot for me so I got to be there 3 years, then another 10 months because I did a medical stabilization. I can't remember the MILPER for the medical stabilization but it starts from the date the dependent is enrolled in EFMP. It's up to 4 years - I got 4 years but they started that from August 2014 - when she was enrolled in EFMP. That's how I got another 10 months after my 3 years was up at my last unit.

Yes, but that determination can lead to a hardship discharge. Many commanders would interpret that if the dependent is so bad off and relying upon you that you can not do your job by deploying to do what you have trained to do that you need to find something else to do besides the Army.