If I buy FCPX for my Mac Mini, can I install it on my Daughter Macbook Air?

LLegally not, if she controls the machine.

If she's at home and you sign on to Home Sharing with your account on her MBA then it can be installed there.

Family Sharing may apply, and if that is the case she can legally use FCP X under your license.

One user (you) sets up Family Sharing, and invites other family members. I don't know if it is required that your daughter lives with you or not.

I do know that Family Sharing allows up to six family members and that they can all legally use the purchases by the family organizer (you).