With regard to your first question, honestly I can't even get my head around answering it but thinking about it made me ask more questions..
1. Wouldn't this only be the case if you were exactly on the axis of rotation of the earth?
2. What are the effects if you fly along the equator and if the answer is yes to your question then surely if you crossed the equator there would be a sudden adverse effects on the helicopters performance?
and finally
3. What sort of a nut job even conceives questions like these?