In the end, is your quesiton: "Should I encrypt my Private Key when I email it to myself, or backup on cloud storage, or some other place?"
Then the answer is yes--absolutely encrypt your PGP Private Key before "backing it up" some place, (encrypt pre-generated Revocation Certificates as well).
An example to create an encrypted ASCII Armor .gpg file, from a plain text Private Key .asc file:
gpg --symmetric --cipher-algo=AES256 -a -o 20141022.PGP.66H049E4.prv.gpg 20141022.PGP.66H049E4.prv.asc
Use a delete utility like, "shred," to delete the original .asc key file, otherwise it can be undeleted.
The Private Key is unusable without the password. However, that Password is a whole lot easier to crack than the Private Key itself.
There are a couple of attack trees employed to take advantage of unencrypted/unsecured Private Keys:
- Brute Force Attack: When trying to "crack" a PGP encrypted email, we don't try to brute force what the PGP Key is--especially if we have a copy of the Private Key on hand. a.) Instead, what we do is subject the Private Key, itself, to a Brute Force attack, to get the password. b.) With that password, we then use the Private Key, to decrypt messages--a far more efficient Attack Tree.
- Impersonation, Harassment: An Unencrypted Private Key can be "Imported" into a PGP store, (GPG, Kleopatra, Windows, etc. a.) This creates the appearance that a particular person used a specific device. In Forensic Information Investigation, a private key on a loaded on a machine is pretty compelling proof, but not dispositive, that a person used a device; In Addition, b.) Public keys can be exported, without the password, (but Revocation Keys cannot). In the end, you can successfully simulate a "footprint" of a user on a particular system, spoof emails from them with a public key attached, etc.
Using a strong Private Key password is the best practice, and will thwart intense Brute Force Attacks.