Do any one-key-of-many cryptographic schemes exist?
Yes, absolutely.  Here is the standard construction to address this problem.  Let $pk_1,\dots,pk_n$ be the public keys of the $n$ recipients.  We pick a random symmetric key $k$, encrypt the message $m$ (using authenticated encryption) under key $k$ to get $c=AE_k(m)$, and then encrypt $k$ under each of the public keys. Finally, we form the whole ciphertext as

$$E_{pk_1}(k),E_{pk_2}(k),\dots,E_{pk_n}(k),c.$$

That's what we treat as our ciphertext.  Now any of the $n$ recipients can decrypt (they can recover $k$ using their private key, and then use that to decrypt $c$), but no one else can.

The same method works in the shared-key (symmetric-key) setting as well.

This standard construction is standardized in more detail in "RFC 4880: OpenPGP Message Format" and in PKCS#7 (enveloped-data).  (Thanks, David Cary and DrLecter!)

You might also enjoy looking into broadcast encryption, which solves a related problem.  It basically allows you to transmit a message to millions of receives, without having a huge blowup in the size of the ciphertext (so it achieves much shorter ciphertexts than the simple scheme above), though there are some details and a little bit of additional coordination required.

For digital signatures, you might enjoy looking at ring signatures and group signatures.