Do parts of a hash carry the properties of the entire hash? When I need to generate unique id's based on some information hashing is typical choice. However, sometimes that id needs to be of a particular size.  I've seen a lot of schemes (HMAC-MD5-96 in SSH, CGA in SeND for IPv6) that use a portion of a hash, so I'm thinking it might be alright to use it that way.  

Which properties of a hash are preserved and which do not apply anymore?

Obviously, with less bits, the chance of collision goes up, but can I still rely on the partial hash for uniform character distribution?  What about the avalanche property?  I'm guessing that if a hash modifies the entire hash upon the smallest change to the input, that would imply that a portion of that hash would be equally well changed.  Am I thinking about this correctly?

Also, would these behaviors be different for different hashing algorithms?
Yes, starting from a appropriately good hash or MAC, removing output bits demonstrably does not weaken it beyond the limit imposed by the number of remaining bits, with regards to collision-resistance and preimage-resistance, and other useful properties (such as impossibility to guess anything on the output without knowledge of the input). Put simply, if a standard hash is entirely unbroken, its restriction is unbroken.

More precisely, one appropriately good condition is secure in the Random Oracle Model. For a MAC, it is also enough that without the key, no attacker could harness enough computing power to distinguish the function from a random function with odds better than 50%, given access to an unlimited supply of $(input,output)$ pairs with $input$ iteratively chosen. For a Merkle–Damgård hash like $SHA256$ there are additional technicalities [we replace the key with the constants in the definition of the hash, and posit that they have been chosen at random; also we must make an exception for the length-extension property: from the length of a message $M$, it is trivial to construct extensions $E$ so that $H(M||E)$ is efficiently computed from $H(M)$ without knowledge of $M$].

Proof: If a function is secure in the Random Oracle Model or/and computationally indistinguishable from a random function, then a restriction with less output bit is also secure in the Random Oracle Model or/and computationally indistinguishable, because removing bits deprives an attacker of information and can only increase the difficulty of a distinguisher. Being secure in the Random Oracle Model or/and computationally indistinguishable implies collision-resistance and preimage-resistance within the limits of the number of bits in the output, both for the original function and its restriction.

It is not appropriately good that breaking collision-resistance or/and preimage-resistance of the original hash or MAC is computationally hard. For example consider $H(M) = C||SHA256(M)$ for some 256-bit constant $C$. This 512-bit hash is at least as collision-resistant and preimage-resistant as $SHA256$, which is rock solid. Yet a restriction to its left 256 bits is the constant $C$, and offers no security at all.