Can someone tell me the name of hondas Motorsports division and have they got a logo? Nissan have nismo, subaru have prodrive, who do honda have?

It's just Honda Racing http://racing.honda.com/
Or do you mean a division that sells performance parts for consumers? If you mean that, then Mugen is the closest. It was started by the son of the owner of Honda and some other guy, but the companies aren't directly related. Though they do are the first to get new cars before they come out, so they can make parts.

http://world.honda.com/motorsports/

The real home of Honda racing.