I understand that a diminished 7th chord is defined by the intervals 1-b3-b5-bb7 and a half-diminished 7th chord is 1-b3-b5-b7, but how do you determine what intervals diminished (and half-diminished) 9th, 11th, and 13th chords have?
The extensions are not touched when dealing with diminished chords unless noted. All extensions can be played as is with the only exception is the 13th chord which cannot coexist with a fully diminished 7th chord since the lowered 7th exists in the space of the 13th.
To make it easier, looking at how it is sometimes notated may help. The Jazz way to write half diminished chords with extinctions is to notate it as a minor chord of whatever extension with a flat 5. For example:
A diminished chord itself has a nice property of being completely symmetrical In most cases you would not use an extension with a fully diminished chord because of the symmetric nature of the chord would be destroyed. You also cannot use a 13th because it is eharmonic with the diminished 7th. However, if you really wanted to you could notate it as a minor 6th chord of whatever extension with a flat 5. For example:
These chords do not come up much in music so the names of these chords are not the best. Also both examples above use more of a "jazz approach" to naming chords. Personally, I would notate the chord you want with putting the diminished sign and then the extensions (C°9 or Cø9).
The intervals are determined by the corresponding chord scales. However, since the chord scale is not unique, there are always several options. For a half-diminished chord a common chord scale is locrian. From locrian you get b9, 11, and b13 as tensions. You could also use locrian ♮2, which would give you 9, 11, and b13. A common chord scale for a diminished chord is the diminished (whole-half) scale, which gives you 9, 11, and b13. The tensions you mention in your comment (b9, b11) are very uncommon for a diminished chord.
- Diminished 9th: 1-♭3-♭5-♭♭7-2 (Ex. Cdim9 will have C-E♭-G♭-B♭♭-D)
- Diminished 11th: 1-♭3-♭5-♭♭7-2-4 (Ex. Cdim11 will have C-E♭-G♭-B♭♭-D-F)
- Diminished 13th: 1-♭3-♭5-♭♭7-2-4-6 (Ex. Cdim13 will have C-E♭-G♭-B♭♭-D-F-A)
- Half-diminished 9th: 1-♭3-♭5-♭7-2 (Ex. Cm9♭5 will have C-E♭-G♭-B♭-D)
- Half-diminished 11th: 1-♭3-♭5-♭7-2-4 (Ex. Cm11♭5 will have C-E♭-G♭-B♭-D-F)
- Half-diminished 13th: 1-♭3-♭5-♭7-2-4-6 (Ex. Cm13♭5 will have C-E♭-G♭-B♭-D-F-A)