As usually, it depends on the context. Without context, leaving out any note of a chord will often make it ambiguous. However, in context, you can usually get away with leaving out a note of a four-part chord. You're right that if the fifth isn't perfect, it is essential for the chord quality. However, I've often found that it can be left out, e.g., in a II-V-I progression in minor, the II chord can be played as a three-part chord (root, third, seventh), and the diminished fifth will be implied. Also the root is sometimes left out, which leaves you with a minor chord a minor third higher than the (left out) root of the half-diminished chord. In context, the root can be implied.
I haven't come across examples where the third or the seventh of a half-diminished chord are left out. Leaving out the seventh reduces the chord to a diminished triad, which can stand on its own, so the seventh can't be implied that easily. Note that the third and the seventh are usually important in voice leading, so it's less attractive to leave them out. E.g., in a II-V-I progression, the third of the II chord (half-diminished in a minor key) usually becomes the seventh of the V chord, and the seventh of the II chord (usually) leads to the third of the V chord.