You understood the private key, public address and sign message totally inaccurate.
Yes that is true that private key is needed to sign a message, but Redoubt should know that the private key is not included while signing, what that are included are just 1. The bitcoin address for signing the message 2.
Private key -- create public key -- create public address
In order to sign a bitcoin message, you need to have a private key for a public address from which you want to sign a message. In a reverse way, the sign message is to show you are the owner of that address by owning its private key.
The message that will be signed, after these has been filled, the user can click on sign message, the message will be signed.
It will require you to enter a password of your wallet. This step is mandatory if you encrypted your wallet. It is a prevention to not allow hacker to be freely sign any message when he find your bitcoin wallet file.
Not all wallets sign a message, not all private keys wallets also sign a message, there are few ones that do it, like electrum and coinomi.
It is not correct (but I don't use Coinomi so I don't know about the wallet). To sign a message, you need a wallet software that support the sign/ verify message feature. The incorrectness is from
not all private keys wallets also sign a message, you can import that private key to other wallet softwares to sign a message.
Say not all private key can sign a message, it is totally not correct.
All wallets signing a message are altcoins private key wallets while not all private key wallets can sign a message, only few ones do.
Why altcoins?