Why does F3D only show part of my FreeCAD assembly when I open the FCStd file?

Yeah, that matches what I've seen. The FreeCAD reader in F3D only pulls in what's visible in the document, so anything you've toggled off in the tree gets skipped entirely. That would explain why you're only getting one or two bodies out of your gearbox, since you said some of the parts are hidden. Turn everything back on, save the file, and reopen it in F3D. That alone should get you the whole assembly.

Tried this on my own file last night, a little bracket with a few copies of the same hole pattern I made with the mirror tool. The copies came through but they all looked fused into one lumpy piece, like the viewer treats them as a single blob instead of seperate bits. Is that just how it handles those copied parts, or is there a setting somewhere I'm missing?

Colors didn't survive the trip for me either, everything came in flat gray no matter what I'd assigned in the tree. Doesn't seem like there's much to do about it yet, might just be how the reader works right now. I've had decent luck exporting a STEP out of FreeCAD and opening that instead when I care how it looks, though you lose the convenience of just pointing F3D at the working file. Curious if anyone's seeing anything different on a newer build, since I'm still a version or two behind.

That matches how I've always handled this stuff anyway. Back when I was shuttling files between CAD packages over IGES and DXF, I learned early to keep a separate 'export copy' of the doc with everythign shown and stripped down for handoff. Same trick works here, make a copy, turn all the parts on in the copy, save it just for viewing. One other thing worth checking, if some of your parts live inside a container or a group in the tree, hiding the parent hides the kids too even if the kids still say visible. I've chased that one more than once.