1) 'I had checked and agree with she should take the responsible.' This sentence is wrong-- it should be 'I checked, and agree that she should take the responsibility.' He checked in the past and now agrees in the present. No problem.
2) 'When to use had had?' Use the past perfect when it is necessary to make it clear that one event in the past happened before a
Good! It would also be a good idea to register here, so that we can get to know you better when you are asking more questions. I look forward to more of your posts.