In a C9, there is the 1st, 3rd, 5th, 7th, and 9th degree in the C major scale despite not having a 7 in the chord. Is the same true for chords with sharp or flat degrees, such as a C ♯9? Would that have a 7 in it, or would I have to classify it as a C7 ♯9?
All 9th chords imply the inclusion of a flat seventh. This includes sharp and flat 9th chords (#9 and b9).
However, there is a good reason why the 7 would usually be included in sharp and flat 9th chord names. It is because it may be unclear whether the sharp or flat symbol is "attached" to the 9 or the letter name of the chord. A couple of examples will suffice to show this confusion:
C#9 would be expected to be the name for the notes C#, E#, G#, B, D#. In other words a dominant 9th chord with root C#. To get the notes C, E, G, Bb, D# you should write C7#9
Cb9 would be expected to be the name for the notes Cb, Eb, Gb, Bbb, Db. In other words a dominant 9th chord with root Cb. To get the notes C, E, G, Bb, Db you should write C7b9
By default, any chord with an "extension" higher than 7 includes that lowered 7th. So C9, C11, and C13 all have a B♭ implied in the chord. It doesn't matter what the accidental is on the 9, 11, or 13, the 7 will always be ♭7.
(Thus your first sentence is technically incorrect; the 7th is not the 7th degree of the C major scale, it's actually the lowered 7th degree of the C major scale. This is because these are all dominant-functioning chords, so they're really acting in the key of which C is the dominant. This key is F, which has a B♭ in the key signature, hence having to lower the seventh from B♮ to B♭.)
- C9: C, E, G, B♭, D
- C ♭9: C, E, G, B♭, D♭
- C ♯11: C, E, (G), B♭, F♯ [in this case, the G is often omitted so as not to clash with the F♯]
Note that "C ♯11" and "C♯ 11" are two different animals. The first is a C7 chord with a ♯11 added, the second is a C♯ chord with a 7 and 11 added. Make sure you're aware which is which. In other words, make sure you know if the accidental applies to the chordal root or to the extension.
The question is not crystal clear to me. Do you mean 'does it HAVE to have a seventh in it', or 'why is the seventh actually not in the key'?
Richard answered the first, but be aware that by calling a chord just a 9th will incorporate the flattened 7th and an ordinary 9th. As Richard stated, it's because it becomes the dominant chord in the key of its IV. Thus C9 contains Bb, as the assumed key is F, containing that Bb, not the expected B.
Also, there are major 9ths and minor 9ths, both containing a 9th, obviously, but the 7th part in maj9 is a major 7, while it's a min7 (b7) in the min9.
If you're talking about a 9th chord without the 7th included, then it's 'add9'. Thus Cadd9 will be C,E,G and D. Rather different from Csus2, which omits the E.