Can you send a Bitcoin transaction to a Bitcoin address that does not exist?
Yes, the bitcoin network and blockchain do not attempt to keep track of which addresses exist. The concept of an address existing or not existing is not meaningful. Addresses can exist but be unknown to the blockchain. I can install a wallet on an offline PC and create valid addresses that no one alse knows about.
What happens when I attempt to make a transaction to this address?
It will be processed in the usual way. The "non-existent" address will then have some bitcoin value associated with it in the blockchain.
If the address was invented rather than derived from a private key then that bitcoin value is forever lost - it can be seen but never spent. Unless, someone can guess a private key from which the address can be derived - the design of Bitcoin makes this as close to impossible as can be achieved - the whole integrity of Bitcoin depends on it.
Is the transaction rejected by the network?
If not is this transaction stored in the blockchain?