1. Most aren't east, they all have varying difficulties, but generally, they aren't the easiest thing to complete the moment you get access to it. Also, even if you can do it ASAP, it's probably better not to, since it's best to first get the skills you want to pass down from the parents.
2. Once you have access to it, you have access to it, period. It will never disappear, only once you complete it.
3. There's no deadline for parents to be S supported. Once you hit chapter 13, any S supports you already created will unlock the corresponding paralogue chapters. Any S supports unlocked after chapter 13 will give you access to the paralogue as soon as you leave the support menu after viewing the S conversation.