Yeah, classical (well, OLD classical) music's copyrights (if they ever existed) have long since expired. That's why if you go look at the classical section of a music store, you will find so many different versions of mozart or bethoven or chopin.
Yes that would work. If you pay the pianist under the understanding that what they're doing is a "work for hire", then you would own the copyright to the performance.
There are usually two levels of copyright -- the copyright to the written music, and the copyright to the performance. Old classical music's copyright is now expired, as TC says, so it's in the public domain. The performance, however, usually isn't (i.e., you couldn't use the London Symphony Orchestra's performance of Mozart) but for your own hired performance, you would own the copyright.
What if I have someone compose/record custom music for my film. Do I have to have them fill out Sync and Master Use contracts, or just a Work For Hire contract to cover the composition and the performance/recording?
if its a work for hire you should own all right including the copyright of the finished sheetmusic and orchestration and the preformance. but you should specify this in your work for hire contract as well so that its agreed upon before they even begin. also in case of any discepancies, you have the contract.