The train companies operate trains, not lines. For example; CrossCountry’s service runs on track shared with other passenger companies and freight from Penzance up to Scotland.
The train companies (in general) operate a franchise for a particular region of the railway (including mandatory service to less profitable areas in the network). CrossCountry is a quirk in that it operates the "Cross Country" franchise and it has no specific track to itself. Eurostar, Grand Central and Heathrow Express are Open Access operators in that they have no franchise of their own but pay to use the track for each journey. The whole thing is needlessly complicated to be honest, there's no real exclusivity - but there's massive capital costs involved in setting up a competitor and more risk involved for these Open Access operators.
Great Western Railway operates the Great Western franchise which incorporates much of south wales and south west of england - Including the Great western mainline. Abellio Greater Anglia operates most services on the Great Eastern Mainline
As for permissions I don't know if the crossrail core section has exclusivity granted to TFL (or its subcontractor MTR Corporation), given it's run as a concession this may be the case.
