The one that stands out for you is VIM.

A. Syntax highlighting (that supports or can be extented to support KiXtart).
Yup - written by yours truly. The syntax file is out of date but I can't say I notice it in my daily use
B. Highlighted text may be exported somehow (copy HTML to clipboard, save as RTF file, etc.)
Save as HTML
C. Windows and Linux program versions. Preferably with one license, if not free.
Native versions for both OSs (and a few more), no charge
D. Smaller and less intrusive into the OS is good. Complete portability is better. (I'm very impressed with TextPadPro's ability to run completely from a floppy, if needed)
Small lightweight footprint - GUI version might be tricky on a floppy but see later
E. "Standard" features -- hex editing, columns, unlimited undo/redo, adjustable tab width display, etc.
Most of these, however I don't recall if hex editing has been built-in yet. VIM (and vi) use a smart-tag in the text which allows you to define many of the settings per-document. And you will wonder how you ever got along without features like "folds" and the ability to pass bits of the file through external filters
F. Freeware/Careware/Open Software preferred. But inexpensive (especially with combined Windows / Linux licensing) OK.
No charge, as already mentioned.

A couple of other pluses which apply specifically to you.

VI is the editor for Unix/Linux. It is available in every version of the OS that I have used. The other contender is "emacs" which has a devoted fan base of it's own.

VI has an enormously powerful command line which allows you to manipulate the file text very quickly and in surprising ways.

VI has a non-GUI console version which has a very small resource footprint and can be used when Windows or X is not running.

The editor will work with both DOS and Unix end-of-line styles and preserve whichever is in use when saving.

The best version IMO is VIM, which you can find at www.vim.org