1) Ordinarily the avios balance of an individual member will expire after 36 months of inactivity but as Avios are deducted pro-rata from each account when you do a reward booking that will count as activity and reset the 36 month counter. Also if anyone in the household account is Bronze, Silver or Gold status that has earned or redeemed Avios within the last 36 months then the avios balance of other Household Account members will be protected from expiry. As your profile indicates BA Silver status your children's Avios should be safe from expiry.

2) I am not sure if the redeposit from a cancelled RFS will count as activity as it's not new Avios but the initial redemption will reset the counter. ‘Activity’ means either earning or spending – on the account.