Question: 1. The time children spend (or spends) watching TV.
2. The time children spend (or spends) working prevent (or prevents) then from focusing on their school work.

Could you please help on the above sentence corrections. And the explanation as well.

Answer: 
It's used to more clearly differentiate between the two meanings of writing for a magazine. It can be both the physical act of writing (pen on paper, on a computer etc.) or it can also mean the act of being employed by the magazine submitting articles sometimes. 

By writing it this way, it's shown that we're talking about her being employed by the magazine, and not that she is writing physically while jack is talking. (presumably taking notes about jacks story)
It's perfectly good to write "She was writing articles for her magazine" which can mean one of two things:

1. At the moment When Jack spoke to her she was actually in the act of writing
2. or, On the day when Jack spoke to her she was habitually writing

The context which follows ("was always looking") makes it clear that it means 2: it describes a habitual action.

"One day, Jack [did something]" is a typical phrase to introduce an instant, at some unspecified time in the past, which is only notable because of what Jack did that day.