Shaun mentions that they did some things to Kellog, explains is prolonged life. Id venture to say that at that point the Minute Men have become a big enough threat to the Institute with you allied to them. Also they could be viewed as being more vulnerable and easier to take out than the well armed and organized BOS
To answer your question:
1. Kellog is basically a Synth. I don't know why they didn't just make him a Synth but they probably didn't want to explain why 80 year old Synths are almost as effective as the modern new ones. He's almost completely machine underneath his flesh with a new suit of flesh over the front.
Hence why he doesn't age.
He's more machine now than man.
2. Your wife was killed 60 years ago and you briefly woke up to see it. Then you went back to sleep until the machinery finally defrosted you.
3. The Institute wants the Minutemen gone because an army of Pre-War Revolutionary-themed heroes inspiring the Commonwealth is kind of the OPPOSITE to eradicating all memory of the Pre-Great War Era.
That's a red herring. That child isn't Shaun.
They were ten, after all.
A science experiment which gets explained later.
Basically Father's clone of himself.
A 10 year old synth version of Father.
I'm trying to figure if this makes Father an enormous hypocrite or not.