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.
The extensions on a diminished chord are a diminished chord a whole step away. So, the extensions on a C fully diminished are a D fully diminished. That gives you natural 9, natural 11, flat 13, and major 7. With half diminished, the diatonic extensions as it occurs in the major scale are b9, 11, and b13, but jazz guys usually opt for the sound of the natural 9 as derived from the 6th degree of melodic minor. Also, if you don’t yet understand the use of diminished scale on dominant chords, and where those chords come from, go check it out.
Generally speaking extensions are heard in reference to something that is being extended. Without establishing the base thing then there can be no extensions of it. Hence if you want to extend a diminished chord you will first have to establish it. Generally speaking it is somewhat irrelevant because, say, a Co7 arpeggio followed by a Do7 can isomorphically be thought of as a Cob13 type of arpeggio. C Eb Gb Bbb | D F Ab. It is no different than thinking of Cmaj13 as Cmaj7 + Dmin or Cmaj + Bm7b5.
It doesn't matter how you think about it, what matters is that you have a way to think about it. It's all about conceptualization. The sound is the sound and it doesn't care if you have a name for it or not. The name only helps us as humans deal with it intellectually.
As for the extensions of chords, you don't have a specific way because there is ambiguity. Generally speaking the modes are what are used as the fundamental structure to base naming from along with context.
E.g., a G7b13 chord will assume to be a dominant from the harmonic minor. That is, it comes from the phrygian dominant. But who's to say it could be from the mixo b6. That is, it has a M9 rather than a m9?
You don't know except through context or making it more obvious in the naming: G9b13 then is a more specific and suggests mixo b6 rather than phrygian dominant.
One of the major problems people tend to have is that they ignore context. Context plays a huge role in filling in the blanks. Some things that make no sense outside of context make perfect sense inside the context which they are used. There is a lot of ambiguity in notation precisely because it's meant to be used to reduce complexity.