Ok Kyle let me make a start. I’ll start with freight since it is my main interest in European Railways. First I will give you an overview of the situation and the big players.
Under the aegis of the European Parliament and its executive arm the European Commission, the “First Railway Package” was promulgated. This is a series of laws requiring member countries to separate the physical infrastructure of the Railways from the Operations. Another part of this package required the principal of “Open Access” for freight, of their networks to any operator provided that they can prove their financial backing and operational knowledge, and they have the requisite insurance coverage. This “Package” required each member country to pass laws setting these into their own statutes. The “First Railway Package” was passed by the EU in 2001, nine years later only 6 countries are substantially in compliance. Great Britain, Germany, Netherlands, Sweden, plus Malta and Cyprus (which have no railways). Norway and Switzerland, which are not members of the EU, would also be in compliance if they were members.
The relationships between railway companies in Europe and Asia are governed by an International Treaty known by the acronym COTIF, as modified by the Protocol of Vilnius, the English title is “the Convention concerning the International Carriage by Rail” This covers the Rights of Rail Passengers, Rights of Rail shippers, and every other matter for International shipments of freight and movements of passengers.
The major organization for railroads in the world is the “International Union of Railways” known by its French acronym UIC. European Rail transport statics can be found here.