Hi Ichthus...those are good questions. I pray for other people lots of times - their relationships also. I don't wait til they ask. If you have a "feeling" you should pray for someone specifically, that's God placing them on your heart. He's giving you a burden to lift them up in prayer. He does that for us & will do it more & more often as we are faithful to pray.
I try to pray for my own relationships & my relationship with God on a daily basis. We are told in scripture to "die daily to self".
I don't know if this will help you, but this is how I keep up lots of times when I am praying. It's easy for the mind to wander & before you know it, you're just plain tired & can't remember what you covered. I use my hand as a guide. I'll explain:
Thumb - those closest to me, my family & loved ones. Also when I pray for my Talk Jesus family.
Forefinger (the pointing finger) - I pray for the accusers of the brethren. Those people who have hurt me & even the enemies we have in actual battle (when my son was in Iraq, I prayed for the hearts of the enemy soldiers to be softened to receive God).
Middle finger - it stands the tallest so I pray for those in leadership - of our country & down to your own city. Also pray for pastors here.
Rind finger - the weakest finger. I pray for those who are sick & those in the care of others. Also pray for children & families in general here.
Pinkie - the one farthest away. This is where I pray for myself & my relationships.
I found this guide several years ago & it has been most helpful. I'm sorry, I don't remember who to credit for coming up with it.