I've signed the Rental Agreement and paid the deposit, but can I move in earlier or later than then the original start date?

For moving in earlier, it is generally no. Once the Rental Agreement is signed and the deposit is paid, the property has been secured off the market and preparations have begun to have the property ready on a specific date.

For moving in later, yes, you can physically move in later than your Rental Agreement start date, but you will still be responsible for paying rent as of the original start date.