Update ----
Why did you say 'Damn it, they really ripped off then. "? I may have missed something but the device responds to a I2C scan. Are you sure it isn't working? Read more below.....
The I2C address of the DS3231 is 0x68.
The other address you found is for the on module EEPROM (also I2C).
Unless you need the EEPROM you can ignore it.
If you were an Arduino hobbyist you would learn the ins and outs of purchasing electronic module. But since this is a one off college project none of that matters.
I've not used a library for my DS3231 but when I look in the Arduino library manager I find the first entry is: this library.
I'm using arduino V2 beta so I don't know if it is in the IDE V1.xxx
I would load it and run one of their examples. I believe this is the fastest way to get things going.
Regarding the charging and the diode. I would recommend you start without the battery at all. Just to get going.
If you then want to use a regular battery and if you don't have the proper soldering equipment to unsolder it....simply crush it with a small pliers. Not very elegant but effective and quick.