The quick answer is, "No" - but it requires thinking through what classical music is.
Is it the music of Old Europe? I ask this because the blues, jazz, country, and rock all originated in the United States. If you eliminate not only those genres but all of their influence, what you're left with (in the West) is the music of Old Europe. Is that what classical music is? If so, then classical music only exists as a very large but closed catalogue, handed down through the centuries, beginning about 500 CE and ending, let's arbitrarily say, with the death of Johannes Brahms in 1897.
But some modern composers think classical music doesn't have to be kept in that cage. They bring in other influences, for example indie rock, and electronica, and traditional music from non-European cultures. For these visionaries, classical music continually evolves, and can only be enriched by contact with new ways of making beauty with sound. Some of these composers are discussed here:
I would include some 20th century composers as falling into the same category as the young composers discussed in the above article. The most famous and obvious is Igor Stravinsky.
And what of movie soundtracks? When I asked my daughter if classical music was dead, she said, "No, because we still hear it in movies." I think a lot of us, myself included, would agree with that conclusion. I would merely add that movie soundtracks represent another evolution of classical, indeed another fusion of classical with something else: it's a fusion with cinematography! I think it's crucial, and exciting, to remember that. Movie soundtracks contribute to the telling of a visual story! Is this new? Actually, no. Ballet has a soundtrack, and opera does as well. Yet cinematography differs from ballet, and differs from opera as well. It's the third evolution of musical storytelling! (I consider Broadway musicals to be an offshoot of opera, differing only in superficial ways, driven more by popular taste than by any serious aesthetic philosophy. Thus we have ballet, opera/Broadway, and cinematography.)
So is classical music dead? No, it's forever evolving.
The easy answer to this is no. Classical music originated in Europe, but it spread to the United States quickly (relative to the limitations of travel and communication at the time), notably with the establishment of the New York Philharmonic in 1842. Brahms was not an endpoint at all. Big names like Bela Bartok, Claude Debussy, Gustav Mahler, Sergei Prokofiev, Giacomo Puccini, Maurice Ravel, Camille Saint-Saens, Erik Satie, and Jean Sibelius all came after Brahms. In the United States, classical composers include Samuel Barber, John Cage, Aaron Copland, George Gershwin (who was also a jazz and show tunes composer), and notably still alive are all of these American classical minimalist composers: John Adams, Philip Glass, Steve Reich, Terry Riley, and La Monte Young.
And yes, film music is a branch of classical music. The aforementioned Glass is a prime example of a classical composer who also did many film scores. SP mentioned some, but other notable film composers include John Barry, Carter Burwell, Georges Delerue, Alexandre Desplat, Michael Giacchino, Bernard Herrmann, Joe Hisaishi, Michael Nyman, Alan Silvestri, Max Steiner, and Dimitri Tiomkin. You can see and hear some of their film music regularly performed by symphony orchestras across the world. For example, here is a fun Lang Lang performance of Silvestri with full orchestra: