I have a question regarding orchestrations (or instrumentation? Are they the same thing?). I was listening to Hamilton and really zoned in on the string parts and it made me wonder - When a composer is crafting a new musical don't they usually do it on piano? If so, once it gets to the orchestrator do they say "okay, the violins should be playing this melody at this part" or does the orchestrator create that on their own? If so, aren't they more like a co-composer? Hopefully I've crafted this question in an understandable way.
It varies widely. Both situations you've described have happened, and everything in between. On the case of Hamilton, Lin is (obviously) very involved in the orchestration process, as are many other composers. Some, however, deliver little more than a melody and lyrics and everything else (sometimes even harmonic structure) are the work of the orchestrator. I would say the former is much more common than the latter.
I'm sure someone else will answer this in more depth, but it depends on the composer. Some composers just write a melody and a basic accompaniment, and then leave the rest up to the orchestrator. I think, in the golden era, there were a handful of go-to orchestrators, which is where the "broadway sound" comes from, because all these big hit shows had much of their main orchestration coming from a handful of people (someone correct me if I'm wrong). Some composers go into more detail with their accompaniment, providing notes or writing parts for important orchestration.
They are like a co-composer in many aspects. Everyone knows the story of Tunick sticking a "Bobby Baby" callback into Another Hundred People and surprising Sondheim, but there's also the kind of sound that has to be put together to really pull the most emotional impact out of a song. Will there be lots of dry woodwinds? Syrupy strings? The amount of input from the composer and creative team varies from show to show but I'm sure they often collaborate relatively closely.
There's an AOL Build interview with Alex Lacamoire, the music director of Hamiliton who orchestrated the show. It's a full 35 minutes of him in front of a piano talking about musical arrangement. He answers that on the original demo for Hamilton's title song, it was just the melody, but Lacamoire added harmonics and built into a string quartet. I really recommend the interview, it's very interesting if you're into music or orchestrations.
Lin is basically piano- and arrangement-illiterate, so he deeply relies on Alex Lacamoire for a lot of the work on his stuff beyond the melody and lyrics (to the point that some call Lacamoire his ghostwriter). I think the term "composer" strictly means music and lyrics so arrangement doesn't count for that.
"Contentment, it seems, simply happens. It appears accompanied by no bravos and no tears."
I think Lin deserves a little more credit than that. He's not great at piano, like many composers. But he co-arranged the show with Alex. Lacamoire puts it perfectly: Lin writes the skeleton of the songs, and Alex writes the DNA.
Another interesting glimpse into how an orchestrator works can be seen in the documentary Broadway Idiot which chronicles the Broadway production of American Idiot. Tom Kitt (of Next to Normal & If/Then fame) took the original Green Day concept album and developed it into something that would work for a broadway cast. The documentary has a memorable scene when they first perform some of the re-orchestrated songs for Green Day's approval.