The "R3" is a version.
I hope you have a Arduino Uno (version R3), that is the most common Arduino.
The outputs and sensors can be handled by the Arduino Uno.
The DS1307 is a very common chip for time and date.
Also the relay shield is okay, but you need 12V. There are also relays for 5V.
To be able to dim the leds, you have to know what is inside.
Is it only leds, or also resistors and capacitors ?
Are those rapidled dimmers compatible with the skkye light leds ?
Normally the Arduino uses PWM to dim leds.
But you have to be sure that the leds can handle a PWM frequency of about 450Hz.