I am not sure this will help, but do you want:

digitalWrite (mosfetLED, Output);     // Output variable modulates LED brightness

OR

digitalWrite(mosfetLED, HIGH);   // sets the LED on >>>>>>>>>>>>  http://arduino.cc/en/Reference/DigitalWrite

OR

analogWrite(mosfetLED, val);     //where val is 0 to 255  >>>>>>> http://arduino.cc/en/Reference/AnalogWrite 

  lastButtonState1 = reading1;     //<<<<<<<<<<<<<<<<<<<< This happens every time you go through loop
  lastButtonState2 = reading2;     //<<<<<<<<<<<<<<<<<<<< This happens every time you go through loop

Perhaps they should be made equal only when something is detected.

Good for you for starting to learn the Arduino platform!

Note: when you compile the code you attached, there are numerous errors in the bottom of your IDE window.
Review these errors
Track these errors down one by one and correct them.
HINT: perhaps you failed to declare some variables.

Get your code to work using 1 button.

Finally add code for the 2nd button.

If I understand you correctly, you can use this circuit. To achieve what you want.

As long as the PWRIN is 1 Volt more positive than +Battery, no power would be consumed from the battery until PWRIN disappears.

I would highly recommend you checkout/buy this book if you want to study the Arduino platform:

http://shop.oreilly.com/product/9780596802486.do?sortby=bestSellers

can some tell me how did we assigned this address:#define DS1307_I2C_ADDRESS 0x68  . how we get this I2c address

This is the address the manufacturer of the 1307 has been given for the chip.

Edit:  Examples here  http://www.ladyada.net/library/i2caddr.html

Common Collector & Common Emitter

inverted signal?

Yes.  Not a problem, use software to invert the results.

http://hyperphysics.phy-astr.gsu.edu/hbase/electronic/transwitch.html#c4

I would use this layout instead:

Try a 12V Window Crank Motor. Inexpensive, low current readily available.

http://www.princessauto.com/pal/product/8366478/
Or
Motor

Edit: the ones I use, form PA, draw 800ma at 5VDC (~17RPM)  and 1amp at 9VDC (~25RPM)
The shaft is 1/2", and for what it's worth I cannot stall the motor with 4" pliers.

You could use Tone.
Frequency could be adjusted from your ping sensor distance calculations.
 
Syntax
tone(pin, frequency)
tone(pin, frequency, duration)

http://arduino.cc/en/Reference/Tone
http://arduino.cc/en/Tutorial/Tone

MOSFET STP60NE06

This is not a bipolar transistor like the TIP120 is, but it will work in place of it.
See the link below for some theory:
http://www.electronics-tutorials.ws/transistor/tran_7.html
And:
https://www.youtube.com/watch?v=Te5YYVZiOKs

You may want to look at a linear actuator device attached to your motor.
http://en.wikipedia.org/wiki/Linear_actuator

This is what you want, excluding the Arduino cctry.

I want to  send a command to my program ,so an hour or a day is added to timer.

When you detect the command you can use the following:

time_t t  = now();
setTime(t + X );                     //X would be 3600 for 1 hour or 86400 for 1 day

OR

adjustTime(X) ;                     //X is + or - moves the time ahead or back X seconds ex: +3600 or -3600

What part do you need help with?