Are there fees for international flights crossing through another country's airspace?
Most countries have implemented route charges billed to aircrafts overflying their territory and by doing so using their control infrastructure. Those fees are usually computed according the distance flown and aircraft weight, small aircrafts (eg. less than 2,000 kg in Europe) may be exempt. They are collected for every flight over the territory, both domestic and international.

In the same way, some countries charge terminal fees when the airport control infrastructure is used.

Those fees are most usually levied for flight performed according IFR (Instrument Flight Rules), not VFR (Visual Flight Rules).

In Europe, the EUROCONTROL organization is empowered by States to collect route charges and terminal fees.