An UUID provides (almost) 128 bits of uniqueness. You may shorten it to 16 binary bytes, or 22 base64-encoded characters. I wouldn't recommend removing any part of a UUID, otherwise, it just loses its sense. UUIDs were designed so that all the 128 bits have meaning. If you want less than that, you should use some other schema.
For example, if you could guarantee that only version 4 UUIDs are used, then you could take just the first 32 bits, or just the last 32 bits. You lose uniqueness, but you have pretty random numbers. Just avoid the bits that are fixed (version and variant).
But if you can't guarantee that, you will have real problems. For version 1 UUIDs, the first bits will not be unique for UUIDs generated in the same day, and the last bits will not be unique for UUIDs generated in the same system. Even if you CRC the UUID, it is not guaranteed that you will have 16 or 32 bits of uniqueness.
In this case, just use some other scheme. Generate a 32-bit random number using the system random number generator and use that as your unique ID. Don't rely on UUIDs if you intend on stripping its length.