As regards slots, there is a long-running debate over who actually owns them at airports in the EU. So far it looks like the airlines, since they have successfully gotten around the rules to be able to "buy and sell" which, in theory, is not possible.
What is clear is that airport slots have nothing to do with bilaterals. For example, bmi british midland has slots at Heathrow which it could use to operate to the USA, but the bilaterals currently prohibit this.
The EU skies are now completely deregulated, as in the USA. Any carrier from any country can operate domestic services within any other EU member state - but only routes within and between EU states. When it comes to operating outside of the EU, the traditional bilaterals demanding that the carrier must be majoritively owned by nationals of the country from which the route is operated remains. For example, British Airways, if it so wishes, could fly between Frankfurt and Munich in Germany. It could also fly between Frankfurt in Germany and Rome in Italy (there is no requirement for either route to go anywhere near the UK). However, due to bilateral constraints, British Airways cannot fly from Frankfurt or Rome to New York.