If you want to use this in a conversation, there would be no need for the subject わたし　since it will be understood from the context. As for placing it, わたしは子供のとき、よくハンバーガーを食べました。　is also fine. Think of it as two clauses, ( I often ate hamburgers) and ( When I was a child ) You can't use two わたし's even though it's possible in English and Norwegian.

It's a common structure in Japanese sentences as far as I know to use the time before everything else, but it is flexible as Yu has mentioned. その　あと、(あなたは) えいが　に　いきますか？ After that, are (you) going to (watch a) movie? If you are going to put the subject there, the way I'm taught, I favour to put the subject before えいが and after その　あと Because if you put the あなたは in front, the sentence will feel more like: Are you, after that, going to watch a movie? Which thinking about it in English also feels a bit awkward, isn't it?