Concerned, yes, but the fear that Palpatine plays up on is his fear of Padme dying, not any concern for the children. Children or no, she was his obsession.
Re: R2's rockets: between the PT and the OT, a flying R2 unit caused a woman to spill hot coffee in her lap, causing an instant lawsuit that led to all R2 units having their rockets ripped out. Simply really. Re: 4, 5, 1, 2, 3, 6 viewing order. I don't think preserving the "Leia is my sister"...