I remembered them (there are loud pops where the edits are obviously very sloppy,
This is a great example of pre-pan pot stereo, with the drums and rhythm in the right channel, and the harmonies in the left. What I heard were Cass's "oh"s punched into the mix. There was a technique they called "punch-ins," where the reel to reel is playing back, and when they got to a certain point, they'd hit "record" on one track for a brief second, then un-punch. The punch could cause a pop. That's what I think you're hearing, and because it's only in the left channel in stereo, it's more obvious than when it's mixed to mono.
I'm still trying to figure out how they created the "player piano" effect. Someone said they put thumbtacks on the piano hammers so they hit the strings in a more percussive way. It sounds a little sped up to me as well, and maybe even double-tracked.
To me, that sounds somewhat more like a muted trumpet than it does an oboe.
There are moments when it sounds like some type of keyboard, like a mellotron. There are no horns listed in the credits for that song on the album, since there are no horns on the album version. So it's hard to say.