Is working as an 'Independent Contractor' under F1-CPT period a problem. please explain

You must be authorized to accept employment. Most student visas do not include employment authorization.

Your question is not very clear. If your work was part of your approved CPT, then it would be fine. But if it's unrelated to your CPT, then it constitutes unauthorized employment. And yes, any kind of unauthorized employment would be considered your fault. Before you assume any kind of work during F-1 status, USCIS expects you to talk to your international student advisor to make sure it is not unauthorized work.

When they approved your CPT they should have defined the conditions of your employment. As long as you are within the parameters of those conditions you should be OK.