If I have both an encoded message and its original, is it possible to infer the private key?
No. that is the point of asymmetric keys (public and private) as you can easily encode a message with the public key (that's what it is for) you have the de and encrypted message. But decrypting the encrypted message is only possible with the private key. The private key can not be regenerated. 


Since it has not been mentioned: this is known as a knownplaintext attack, and would be a very bad vulnerability for a cipher to have. In fact, in publickey crypto this attack is equivalent to a chosenplaintext attack, since anyone can encrypt any message! There have in fact been some known/chosenplaintext attacks against RSA (the publickey cipher most often used by PGP to encrypt the symmetrickey), but they usually involve sidechannel attacks against poor implementations, or require an insane number of ciphertexts, or only work against certain, rare keys. However, in general, the answer to your question is "no"  that is what makes the cipher secure. 


No this would be very bad. PGP encryption uses a serial combination of hashing, data compression, symmetrickey cryptography and finally publickey cryptography. Hashing makes use of oneway functions. A oneway function is a function that is easy to compute, but hard to invert. Publickey cryptography makes use of asymmetric algorithms.
Just having the original message is not enough to infer the private key. 


There are two answers to this. Is it theoretically possible? Yes. You could generate a fake key, try the encryption, see if it matched. If not, try another fake key, encrypt, see if the output matched. This is commonly called a brute force attack In practice, this can't be done. It takes a nonzero amount of time and computing power to try this. PGP encryption would usually use a decent size key, maybe 512, 1024, 2048, or even more bits. The keyspace (the number of keys you could possibly have) is so big that it would take you forever to find this combination. My guess is you were asking if you could really do this, not the theory. And so the answer is no. The world doesn't have enough computing power for this. If you asked on security.stackexchange.com someone better familiar with the algorithms and math could tell you how long this would likely take (possibly until the sun metastasizes and boils our planet). My short answer is "not within your lifetime" 


Not likely, it might help but it should not get you too far. The security form is probably a better place for this, but: You should have multiple steps to the encryption, at minimum even a decent should have a seed, a hash and a salt. For example, you might have a salt to make all passwords 64 bits long, and then run that through the hash. If it were only a hash, then it would be very easy to crack. This is a very simple version of it... but it is not that easy. 

