For instance, if I am trying to send 1 BTC to an address, however, I do not have enough BTC. What will happen to this transaction?
Will the sender and the recipient both receive notification regarding the status of transaction?
Unless you are using modified software, your software will not allow you to create such a transaction.
Even if you succeeded at creating it, the transaction is invalid. Invalid transactions don't get relayed on the network, so your recipient would never learn about it.