I'm looking for some free editor for Windows that I can use to adjust levels and curves on 16-bit TIF files. Do GIMP and/or Paint.NET support 16-bit depth? What are my options?
I believe Paint.NET is limited to 8-bits per channle but I don't have a reference for that.
Mainstream Gimp is limited to 8-bits per channel.
Variants of Gimp used by the film industry have greater bit-depth
Film Gimp has vast dynamic range: 16-bits per channel (64-bit RGBA)
Film Gimp is also known as CinePaint
GIMP version 2.10 will will support bit depths higher than 8 bits including 16-bit integers and 32-bit floats. This version could be probably available in one or two years.
Currently you can test the functionality in the development branch of Gimp 2.9. For Windows you can get the development builds here: Nightly Builds of The GIMP.
Check out ImageJ from Nat Inst of Health (USA). Java client program that is very powerful but not super user friendly.