How to make a browser download "Always start here"

Can Windows experts save & print always starting from a defined folder?

FireFox Tools,Options,Downloads has only two download options:
- Ask me where to save every file
- Save all files to this folder [folder-name]

What is missing is an "Always Start Here" option:
- Always start here [folder-name] & then ask me where to save every
file

Likewise with printing to PDF. Is there a PDF printer that will set the
default folder yet still have the option to click about in the
hierarchy below (or above) that folder to save a file?

This "Always start here" capability would be a godsend - if only an
expert out there could tell us how to accomplish this (a decent
workaround could be acceptable if it was better than what we have
today).

Any suggestions?
Susan

Note: I'm aware that each time we save any particular file type, the
next time we save that same file type, the save dialog starts deeply in
the hierarchy depending on the file type where we last saved a similar
file type. When you save mixed file types, you get a haphazard set of
starting directories instead of a single starting directory from which
you can then navigate downward (typically, but upward should also be
possible).