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Let's say address X has 1 btc. Can you create a transaction which would send the 1 btc from X back to X and broadcast it without having to sign it, since the 1 btc is going back into the address it is being sent from?

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Transactions don't send from address to address.

Transactions spend specific coins ("transaction outputs") and create new coins as a result. Every coin has an amount and an owner (identified by its address).

While it's possible to define the "balance" of an address as the sum of the values of all coins with that address as owner, this is not how the system works internally. Addresses are just an authentication mechanism to make sure coins can't be spent without proving ownership. In particular, a transaction that sends coins "back to the same address" is in no way different from one that doesn't. It's still spending a coin and creating a new one - just one that happens to have the same owner. In order to spend a coin you need to prove you own it.

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Unsigned transactions are invalid and will be rejected by all miners and other nodes.

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