1. Potentially 3 ways. A. Most long haul aircraft have sat phones so they can talk with their operations people. It’s required under FAA regulations for aircraft operating on longer flights when more than 3 hours away from a suitable airport after having an engine fail. B. Starting around 1980 ACARS (Aircraft Communication Addressing and Reporting System) started showing up in most airliners. Think of it as text messaging the company. C. Really old school but still in place - there are several “Radio” companies. New York radio, Stockholm radio, Tokyo radio, etc. They have VHF and HF radio frequencies separate from ATC frequencies. You can call them up and have them pass a message or if needed get a “phone patch” where the radio operator will call the company operations and then connect the flight crew with their operations. This ties up the frequency so is normally only done when desperate.
2. It could be done through ATC or through the company by the means above. It would probably be a matter of if the crew was concerned about everyone hearing the call. Using the ATC system would probably be faster as they would have the local police number handy.