Yes, you should still tip your server the "recommended" amount for the kids even if they didn't eat with you.
Tips are not pooled. Your server keeps the tips you give him/her. However, as has been stated, the tips do reflect the service he/she gave to SOMEONE at breakfast and lunch, even though it might not have been you. And his income was based on your child's place tipping, even if the chair went un-occupied. It is not customary to tip at breakfast or lunch, although they certainly won't refuse a tip if you leave it. The single tip for the week reflects all the dining room services received for the week (well, 3 tips, to waiter, assistant, and head)
Tips to bar staff (the automatic 15%) are pooled and divided among those persons doing bar duty. It is my understanding that any additional tip you add goes to the person who served you the soda or drink.