In the EU, can I start an R&D company and apply for research grants? I am from an EU country and hold a PhD degree in structural mechanics. I have two questions:

Is it possible to start a company dedicated to R&amp;D only and apply for research grants for my own projects and ideas? I would be the one hiring research staff and I would be supervising their work.

Does anyone know any funding organization, body or foundation that is able to fund privately-held companies?
A significant part of the EU's research framework program (current Horizon 2020) funding is for industrial and technology research, so companies (or consortia which also contain companies) are eligible for funding.  However, the idea here would be to ultimately aim for developing a product - thus, a pure R&amp;D company like what you mention might not be the right target.  (Of course, you can always aim for having a company whose product is intellectual property.)