Notes do not have timbre, they are just named pitches (which themselves are just frequencies).
Hm? If it's about how they appear on the staff, why make the distinction?
There are different ways to assign pitches to notes. What is commonly used in modern western music is called equal temperament, but there are other systems, like just intonation, for example.
The pitches in alternate tuning systems like JI and maqam are sometimes, but not always, denoted with extra information. Quarter-flats, cent indicators, and so on, distinguish the note-pitch representations. While tuning systems can indeed be transposed or converted to unmodified standard (ET) staff notation, that's more of a convenience that has its own learning curve.
So, in JI you wouldn't simply use 'A' to represent (e.g.) 440Hz, it might be A 4, for the "A" key (or "A" staff position) having a 4 cent difference than ET A.[1]
1. http://lilypondblog.org/wp-content/uploads/2015/11/example.p...
Yes they do, in the forms of texture and timbre.