@RedKi-rr
You should have more cities than three by 1000 BC. You need to REX peacefully or settle one to three cities and perform an early rush to get more. Your rush was a reaction to an AI settling where you wanted to settle; it was not a premeditated rush. You need to commit either to peaceful REX or an early rush. Doing both at the same time takes more skill and will often result in failing at both at Deity Level.
The only building that should be built early is a Granary. Building other buildings too early takes wastes hammers that are better spent on Settlers and Workers (if not stolen) for REX or military units for early rushing. This is a common mistake at Deity level. There are some exceptions, like a library in the capital to hasten Horseback Riding in a Horse Archer rush, but these are rare.
You need to adapt to where the AIs settle. If an AI settles too close to where you want, you need to adjust your settling spot or go for an alternative settling spot. Fighting for the contested settling spot rarely works to your advantage, often wasting time and hammers as a result. Faster REX solves this problem more efficiently by settling where you want before the AI can do the same.
Often, you should settle less desireable spots to block the AI from settling near your capital. If two or three blocking cities can settled, you can settle several more cities between your blocking cities and your capital at a more leisurely rate where you can also build more workers (if not stealing enough), library in the capital as well as settlers, though settlers should still be the highest priority.
Don't forget warriors for fog busting though; they can be even more important than settlers throughout your REX phase; not so important as in the early rush, since barbarian units provide experience for your early rushing units to gain that optional 1st (no barracks), 2nd (optional barracks) or even 3rd promotion (barracks and raging barbarians?).
To make things a little easier, select No Barbarians. Then you will not be concerned with fog busting and can focus more on building settlers and send them out without a defending unit.
A leader with the Imperialistic trait can make REX much easier with the +50% hammer bonus for building Settlers. A city with a 6F tile and more food resources plus several grassland mines can become a powerful settler pump without resorting to slavery or forest chopping. Whipping and forest chopping are the fastest ways to build Settlers though, whether you have the Imperialist trait or not. Be sure to maximize hammers to make the most of the Imperialistic trait though.