Looking for nonces of even numbers
I think that Tim S. may have the answer with his comment about endian-ness.  

Your observations about the nonce having its lowest byte zero (being a multiple of 256), are with respect to the little-endian byte order of the block itself.  From the perspective of a big-endian machine, these are statements about the high byte of the nonce.

So consider a miner which is big-endian.  The natural algorithm is "start with nonce=0, compute scrypt, increment nonce, repeat", so your "even" nonces will be tried first.  However, when a new transaction (or a new block from another miner) arrives, a new block header has to be constructed, and it would be natural to restart the nonce at zero when this happens.  In order to get a nonce that is "not a multiple of 256", it has to complete 2^24 hashes before being restarted.

Of course, x86 is the most common desktop CPU, and it is little-endian, but most scrypt mining is done on GPUs.  I hypothesize that a majority of these GPUs are big-endian, or at least that some common mining software causes them to increment their nonce in a big-endian manner.  Does anyone know if this is the case?

From this chart, it looks like modern GPUs can run scrypt at roughly 1 Mhash/sec.  So 2^24 hashes would take 16 seconds.  Litecoin is currently averaging roughly 10K transactions per day, which is an average of one every 8 seconds or so.  So it would not be surprising that a miner would  usually not get into their high byte (which for you is the low byte) before restarting.

This hypothesis would also explain why we do not see such a pattern with Bitcoin.  Current SHA-256 ASIC miners run at many Ghash/sec, and so are very likely to go through all 2^32 possible nonces before being restarted by new transaction data.  (We might see patterns in the extraNonce, though.)