Do you have the little WiFi Module board with the level shifting FETs and 3.3V regulator and such on it or just a bare ESP?
If you have only a bare ESP don't forget that it needs 3.3V not the 5V the clock runs on. Level shifting of SDA and SCL is nice but probably not absolutely necessary. I've found that the rumor of ESP's I/O being 5V tolerant is true. Also something somewhere has to provide pull-up for SDA and SCL. If you have the little module board it handles that for you.
Look into it later when the dust is clearing off the crater.
I got it logged into my network and talking to the timeserver.
However, the ESP8266 does not seem to be talking to the clock. The time never updates, nor do any changes take place made in the settings tab.
IC2 slaves are shown at 87, 104, and 105. I assume this means the ESP can see the clock and the DS3231ok, so not a wiring issue?
I tried updating the firmware on the ESP8266 without much change. The LED on the ESP works as described, but still no communication with the clock. The clock does not seem to be able to read the DS3231, as the time always starts over after I reboot the clock.
That "87" on the I2C bus is bothering me. The RTC is 104 (0x68) and the Atmega is 105 (0x69) but where is 87 coming from? If it's a simple matter to temporarily disconnect the RTC it would be interesting to see if the 87 disappears along with the 104. This may or may not have anything to do with the problem you're having.