SSDI is Social Security Disability Insurance and is solely available to people who have worked 40 consecutive quarters while paying into SS.
Your kids would be looking for SSI, unless you or your spouse are on SSDI now.
Assuming you are not on SSDI, answers would be:
A. Yes and yes
B. Probably
C. When they are 18 and on their own, I think.
Pam R.