I don't know that there's a good consensus for what exactly constitutes a "new work" any within creative field. Obviously, there some clear lines (like straight copying of elements, etc.), but typography raises an interesting problem in that there are only so many variations on the basic form of each letter. For example, there are alternate (two story) versions of the lowercase letters G and A, as well as numerous ligature variation.
Other variations center around the proportions of the letterform. Most alphabets are quite flexible when it comes to modifying character shape: a quick image search will show the variety possible. These considerations can significantly alter the resulting typeface's tone and legibility. OpenDyslexic is a good example of what modifying a letterform can do.
What one person considers a ripoff another could easily consider a poorly done
tribute or redraw. (See the issue of Helvetica vs. Arial for an example of that).
Not to be that guy who is super precise: each and every imaginable variation, regardless of how small the change was, would technically constitute a new font, but I think you're asking about typefaces. See AIGA's They're Not Fonts for a bit more on the difference and why it's very relevant here (it really is, though).
As far the specific font properties you describe (position, slant, width, style of numbers, letters, symbols, and various cases), those have formal definitions that fit within a typeface family.
For example, consider the superfamily Univers.
(source: harsco.com)
As you can see, there are many variations of character width, stroke thickness, slant, and stroke angle — however, each of these variations are considered a single typeface. Thickness is often defined on a scale from ultra thin to ultra black, angle be described with terms like italic (though italic is more than just angle) and oblique, width can be ultra condensed to ultra wide.
Additionally, type families can include variations such as small caps, monospace, cursive, even scientific or mathematical (often containing many specialized symbols for equations or technical diagrams that wouldn't make sense for every day use).
To get a better understanding of how fonts and typefaces fit together and what constitutes a new one of each, I would highly recommend reading Ellen Lupton's book Thinking With Type — the freely available companion website is also a fantastic resource and doesn't require a payment to Amazon et. al.
Bringing all of that back around to answer your question: any of the variations that would fall under under a family or superfamily should be considered part of the original family. An example: you create an extended symbol set and lining figures for the Fanwood typeface. You have simply added to the original family and your additions might be best considered "Fanwood Scientific," "Fanwood Technical," or even "Fanwood Numeric" — they are, however, part of a new font.
However, say you were to find an old manuscript that you wanted to use as the basis for a new typeface, (provided it wasn't a known face already) you could feasibly redraw the letterforms, optimize them for a specific display case such as screen, and consider it a new typeface. Obviously, the ethical thing to do there would be to cite the source of inspiration or reference, but it could be (keeping in mind that "new work" is a fuzzy term) considered to be a new and distinct typeface.
