A 0.1uF capacitor and a 1kΩ resistor in series can be used to create a power-on reset circuit for the DS1306.
The DS1306 has an internal switchover circuit that automatically switches to the battery when the main power fails. No external circuitry is required.
The DS1306 can handle clock frequencies up to 32.768 kHz.
Yes, rechargeable batteries can be used with the DS1306, but the charging circuitry must be designed to prevent overcharging and ensure proper battery health.
The initial time and date can be set by writing to the time and date registers using the I²C interface. A microcontroller or other external device can be used to set the initial values.