Sentences 1, 2, and 4 are yes/no questions, so you have to insert if or whether. And you have to report who said this, unless you want to make these passive (which is a whole different issue).
1. He/She asked (me) if/whether I was/am good at organising people.
2. He/She asked (me) if/whether I enjoy(ed) working in a team.
4. He/She asked (me) if/whether I listen(ed) to other people's advice/suggestions.
The others are wh-questions, so the usual subject-verb inversion doesn't work in reported questions.
3. He/She asked (me) what I do/did when my ideas don't work.
5. He/She asked (me) what I do/did when I have too much work.
6. He/She asked (me) what time I normally start(ed) work.
In all of these I have given the option of making the verb present or past. These are all questions about my habits or character, so they remain true. They can be reported with either verb tense.