In the EU, can I start an R&D company and apply for research grants?
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.)