GPG/PGP Signatures & Encryption - An Academic Security Question

Solution 1:

You are basically asking if applying a signature and running a decryption are performed in the same way - the answer is no.

You wrote:

Digital encryption and signing is basically the mathematical multiplication of a mathematically-suitable representation of a message, by a very large number (the public/private keys). When the private key is "multiplied" by the public key, they cancel each other out (somewhat simplified description).

This is not correct. Signatures are performed on cryptographic hashes of the cleartext, not on the cleartext itself.

Solution 2:

Error in assumption; the opposite of encryption is decryption, and signing does not operate transitively on either. Signing a message has no effect whatsoever on the encryption.

Solution 3:

In principle signing is only done on the hash of a message, not on the message itself.

Besides this, your scenario does not sound very likely at the moment. Nobody provides an automatic signing services. It wat practical case would this really happen? Wouldn't somebody look before they sign something? Why would they sign (appearently) rubbish?