I wanted to extend NumericStepper with a class called TimeStepper. That would
let you step though the minutes of the day but display and read like '5:30 PM'
or '17:00'

Unfortunately, pretty much everything in NumericStepper is private and I need
to do things like allow characters to be entered into the text field.

So I copied the code from NumericStepper.as to my code tree, modified it and
saved it as TimeStepper.as.

My problem is that there are some relative includes:
include "../styles/metadata/styles/metadata/BorderStyles.a s"
include "../styles/metadata/FocusStyles.as"
and
include "../core/Version.as";

How do I make these work? Or for that matter, am I even doing the right thing?

You can copy those includes to the proper relative path as well.

Its always preferrable to just extend an existing class, but if you can't do
what you want by normal extending, ya gotta do what ya gotta do.

Ok, so I copied the referenced files into my src directory. It finds the files and compiles but there are no skins so I can't see the up/down buttons and such.

What is going on?

You probably have to modify project properties to point to the standard Flex skins.