I've got an "XS" Pokémon. Is it special?

There is nothing particularly special about an extra small pokémon. Typically, smaller pokémon appear to start with lower CP values, but that is about it.

That said, there may be further medals that relate to the weight of a pokémon. For example, there is the Youngster medal.

A Pokémon with an XS tag, for either weight or height, is a sign that this Pokémon has lower than average values for those stats. As was explored in this question, it appears that higher weight leads to higher health for two Pokémon of the same CP value. From this, one can conclude that a Pokémon with XL tags for height and weight is generally going to be a better choice than the same Pokémon with XS tags.

I've read in a few places that these tags also influence your Pokémon's moveset, although I haven't seen any evidence to back this up.

Actually, the size has a lot to do with the Pokémon's stats in the game. I leveled up two Pidgeys until they were both at the same point on the CP meter. One was an XL and the other regular. The XL's CP was 79, while the other's was at 73. So imagine the margin between an XL and XS Pokémon. It could be a very big CP difference.

And to the above theory that an XL Pokémon requires more stardust and candy - it's simply not true. I have an XL and regular Venonat at the same spot on the CP meter and they both take 400 stardust and one candy to upgrade.