A schematic of your circuit would be nice to have (you didn't provide one) but since you provided photos, it can pretty much be derived.
Little details really matter.
Not that it matters now, but you had told us that the RTC module contained a LIR2032 but the photos show a module with a CR2032.
Are you really powering your circuit with 4 AA batteries in series like the fritzing diagram is showing?
6v is too much voltage.
Beyond that, the biggest thing I notice is that it seems that what you have hooked up right now is not matching what you previously told us you had connected.
This s/w and h/w is not attempting to turn off the power to the RTC to put it into lower power mode.
And from the photos, there is no wire connecting D7 to the RTC.
In looking at the actual schematic for that RTC module there are some design issues that prevent you from using it "as is" for your application.
The LED (for power), and the charging circuit (which was a bad/broken design) have already been removed but there are some VCC and pullup issues that still need to be resolved.
Here is a link to page with lots information on the subject:https://edwardmallon.wordpress.com/2014/05/21/using-a-cheap-3-ds3231-rtc-at24c32-eeprom-from-ebay/
There is lots information there including a schematic of your RTC module and links to other places with additional information.
It is pretty dense so take your time reading.
The key thing to note is about the pullups.
What complicates things is that they are needed but in your application you are using dual voltages for the RTC module (module VCC and backup battery)
The RTC module has pullups on it for i2c and for SQW. These pullups are connected to the RTC module VCC which creates a problem when you use an arduino pin to power the module since when you turn off the power to the RTC module VCC by grounding the signal (which is required to push the RTC into low power mode), the RTC module pullups will now be pull downs since they are now connected to ground.
The page above talks about this and how you will need to make some additional changes to the RTC module in order to be able to power it down and use the SQW signal.
They are not difficult but are required because the module was not designed to be used in lower power mode.
In your case, you can use the AVR internal pullup so no external pullup will be needed on the SQW pin.
Also, the AVR has pullups on the SDA and SCL pins. While they are WAY out of spec, they often do work and it may be worth a try to see if it works without external pullups on the i2c bus signals.
While it most likely will work ok without external resistors on the I2c bus signals using only the internal AVR pullups, keep in mind that it is out of spec so there maybe issues depending on the main battery level, the length of the wires used, and the temperature, so for maximum reliability, you may want to use a pair of 10k resistors.
You also need to address the transistor that you are using to power your servo.
As I mentioned in response #42, you cannot use a 2222 transistor the way you are trying to use it.
The way you are using it can damage the AVR or the transistor.
A NPN transistor works by turning on a switch between collector and emitter when current flows between the base and the emitter.
The important thing to keep in mind is that NPN transistors are current driven. It takes current into the base to turn on the "switch" and the more current you need to switch between collector and emitter, the more current you have to push into the base but you do have to limit the current. The way you have it wired up now, you are not limiting the current so it is overloading the AVR pin.
You need a resistor between the AVR pin and the base transistor pin a 1k resistor would probably be fine.
Keep in mind that the needed current that flows into the base of the transistor is wasted.
Alternatively you could switch to a FET transistor like 2N7000
These work on voltage rather than current.
When you apply a voltage to the gate pin, it turns on the switch between the drain pin and the source pin.
You hook it up like this:(Device VCC signal connects to VCC)
(Device GND signal)------------+
(Arduino Pin)-------------G-|-< (2N7000 FET)
This works the way you were assuming the 2222 worked but it doesn't use/waste any current so it is more power efficient than the 2222.
Not sure where you get parts from but I've been happy with this place for low volume low cost parts like this:http://www.taydaelectronics.com/
Also, when you add back in the code to turn the power to the RTC module on/off you will need to turn it back on when you wake back up.
The older code that you had was not doing this. While the RTC will work with i2c when running off battery its battery backup power, that isn't what you wanted since I assumed you wanted it to use the main power when being accessed.
Hope that helps.
You are slowly being dragged into the world of having to understand the low level hardware.