Bitcoin P2PKH transaction - two questions
The signature is mathematically generated using the private key and the message, but is not the message itself (it uses the hash of the message during the signing process, the message is not recoverable from just the signature). It's just a value (specifically, it is 2 numbers, called r and s) which can be verified against the public key and the message to see if it's a valid signature. In this case the message is the transaction data, which also protects the transaction from being modified because if it was modified, the signature would become invalid. If you're interested, you can read about the Elliptic Curve Digital Signature Algorithm (ECDSA) here.

Indeed it would be possible to use just the public key without the hashing steps, this is known as Pay to Public Key, P2PK and is also an acceptable output type. Please note, though, that a bitcoin address is not the public key itself. To create this type of output, the sender would have to know the public key, to put it in the output. It is not possible to obtain the public key from a bitcoin address because the address is a hash of that public key, to make it shorter and easier for common use, and is thus a non-injective trapdoor function.