How would you translate: "Это фирменный поезд."? "The firms train"? or privately owned train company.
I'd translate it as "premium train service" or "luxury train service" (or, literally, "signature train service").
That's a legally defined concept in ex-Soviet rail service which designates some popular routes.
Such a route has to have a proper name (like 001/002 Красная Стрела, the first route of that kind), a livery, satisfy certain requirements to the rolling stock etc.
They usually have more convenient arrival and departure times, offer some perks like better linen and snacks, include a restaurant car.
They are of course also usually more expensive than ordinary routes (sometimes slightly, sometimes significantly) within the same car class.
I am a little curious about how come they can offer faster train service. Do they stop at fewer stations and drive faster (have newer train) or perhaps make very short stops at each station?
They make shorter stops on the stations and have priority over other routes (if the line is to be shared by a premium and an ordinary route, the ordinary one has to wait). This means trains on ordinary routes just move slower, as yielding to the premiums is expected.
They also refer in the text about that they greet the train with music. Is this common in Russia? I read somewhere in another text in that music was played when that specific train was leaving its starting point.
Yes, this is quite common on those routes. They do play Gliere's "Hymn to the Great City" when 001/002 Красная Стрела arrives to St. Petersburg, and "Farewell of Slavianka" when 31/32 Тамбов departs from eponymous Tambov.