How do you power your setup? Just by USB? The DS1307 is quite picky about the voltage of the supply power, especially if the module has a fresh backup battery. If the voltage drops below 1.25 times the battery voltage the module stops talking to the MCU. Fresh batteries often provide a voltage of 3.5 to almost 4V. If your USB voltage drops a little bit below 5V (which is often the case) you cannot communicate with the DS1307.
To set the time I conect Arduino to the PC by USB and I put the battery in the RTC. I've got more or less 6V of supply with the Arduino (I don't know why if it is said to be 5V) and 4.48 V in the battery, so at first it doesn't seemed to be the problem.
I don't know...
Are you sure your multimeter is measuring correctly? I think both values are relevantly bigger than usual. Try powering the Arduino with a 9V power supply.
First off, replace the battery in your multimeter. If it reads too high (like it does now)
Second, in Arduino Examples → Ethernet there is a NTP-example thar retrieves the time over the internet, and writes it to the serial terminal. You could easily change the code to suit your needs.