Can you send bitcoins together from the same wallet if they have been received via various different addresses? What has been sent to your individual bitcoin addresses is supposedly recorded in the blockchain, but can you then send all of the combined bitcoins (that you received via various different addresses) at once with your public key? For example, if I received 1 bitcoin via address 1, 2 via address 2 and 5 via address 3, all of these addresses being derived from the same public key, and I wanted to send 6 btc to a friend, could I send all of them at once with my public key? Or would I have to send a combination of bitcoins from different addresses, i.e. by separate transactions? And if I could send them all together using the public key, how is it determined that I am in possession of all of these bitcoins if they were sent not directly to my public key, but to various irreversibly hashed addresses derived from my public key?
A bitcoin transaction can have one or more inputs, and one or more outputs. For example, here is a transaction with 1 input 3 outputs, and here is one with 3 inputs 1 output. 

I think you are confusing the terms public key, and private key. A wallet is a collection of keypairs (private keys and public keys), and each one can be used to create a unique bitcoin address. So if you have three different input transactions to three different bitcoin addresses, each of these will be spendable using it's own private key. 

Your wallet will sign the transaction with each of these three keys, and the rest of the network can validate the signature by comparing it to the public key associated with each input (despite never knowing the private key!). To learn more about how this works, look up some info on how public key cryptography works, it is a useful cryptographic tool employed in many systems, not just bitcoin.