To your last point on Music: There is a Your Lie in April CD album that the Sony has released containing all of the classic pieces of the show as performed by the show's pianists (Tomoki Sakata & Eriko Kawachi) and violinist (Yuna Shinohara). You can hop on YouTube to find their recording of Introduction & Rondo Capriccioso.
You can find the full CD on Amazon, though it comes with a hefty 30 USD price tag as an imported product. And obviously wait until you have finished the show first.
Shigatsu wa Kimi no Uso: Our Music Book is the album you're looking for.
The album contains most of the classical pieces from the show and has multiple performers. Tomoki Sakata being one of them.
You didn't ask for this but there's another album called Shigatsu wa Kimi no Uso: Twinkle Little Star that has other classical pieces. Some of them included in the anime, some weren't.
On top of the answers already provided there are 2 more soundtracks.
Shigatsu wa Kimi no Uso Boku to Kim to no Ongakuchou = all the performances in the show
Shigatsu Wa Kimi no Uso Twinkle Little Star = Other performances related to the show