Does an encrypted message contain any information about whom it is encrypted to, or at least to how many recipients? And if so, is it at least only obtainable for other recipients or for just anybody?
To a large extent, yes. See RFC 4880, section 5.1: for each recipient, there is a "Public-Key Encrypted Session Key Packet" which contains the recipient key ID. This key ID is a 32-bit value which is used as index in key servers; there are key ID collisions, so this is not an absolute, totally accurate indication of the recipient, but it still gives a lot of information.
Theoretically, an all-zero key ID can be used as a way to discourage traffic analysis, but this is not a complete solution. For instance, with RSA keys, the packet still contains an integer between 0 and n-1, with n being the RSA modulus, with a fairly uniform distribution. Thus, observing many messages can yield, statistically, the first (most significant) few bytes of that modulus, which can be used to discriminate recipients from each other.