Assymetric cryptopgraphy and JWT

So I'm trying to understand how JWT is implemented in a codebase that I'm reviewing. I read this and this

However, it seems in this codebase, the client has the private AND public key... It has the public key of the server and its own private key (I assume the server has the corresponding private key). Why is this necessary? Shouldn't the client only need the public key and the server only needs the private key? If the client is encrypting a message, can't it use the public key to encrypt it and the server just needs the private key to decrypt it? Conversely, can't it decrypt encrypted messages from the server with its public key? Why would the client need two sets of public and private keys?

From the reading:

To create a digital signature, signing software (such as an email program) creates a one-way hash of the electronic data to be signed. The user's private key is then used to encrypt the hash, returning a value that is unique to the hashed data. The encrypted hash, along with other information such as the hashing algorithm, forms the digital signature. Any change in the data, even to a single bit, results in a different hash value. This attribute enables others to validate the integrity of the data by using the signer's public key to decrypt the hash. If the decrypted hash matches a second computed hash of the same data, it proves that the data hasn't changed since it was signed.

1. What is the differnce between the hashed data and encrypted data? Why do you need to hash it first? Are hashed data encrypted?

2. Where doe that secon computed hash come from? If the decryptor is attempting to apply the public key to the encrypted data... how does it know it succeeded? What does it compare it to? Where does that comparison hash come from?