i think google adwords has no concern from google organic search please can't you give refrences for what you said this is totally dump question if you have some know how of seo.
there is no reason to believe this. if this was true i could just pay for adwords to improve my ranknings and then just make money of the traffic they are bringing me in for free.
Worked with dozen of clients, building their site from the ground up, and not once I saw google adsense directly affecting their rankings. So, from my experience, the answer is no.
As far as I know it might help your new site get indexed quicker but it has nothing to do with ranking.
I believe YES - however no evidence. After having many types of SEO and Adwords clients I believe there is a correlation.