Electronic – I2C pullup resistors not working in simulation

There is a typo in the Atmel datasheet, the rise time for the 100kHz case should be 1000ns, not 100ns (it would not need to be lower than the 400kHz case of 300ns) then you get:

\$\dfrac{1us}{400pF} = 2.5k\Omega \$; for the 100kHz case

The LCD datasheet (almost certainly) means the maximum bus capacitance, not the capacitance that it adds to the bus. It probably adds around 10pF. You can either check with an LCR meter or just set it up with a 2k resistor and look at the rise times.

Many devices don't fully comply with the official 400kHz specs, so it's best to refer to these for understanding the conditions under which 400kHz can work (bus capacitance, pullup/current source/etc) See section 6 onwards in particular (for example see note 4 on pg.47:

[4] In order to drive full bus load at 400 kHz, 6 mA IOL is required at 0.6 V VOL. Parts not meeting this specification can still function, but not at 400 kHz and 400 pF)

Further on, these tables are pretty helpful, and look to agree with your calculations:

1) What happens when the I2C pullups are omitted?

There will be no communication on the I²C bus. At all. The MCU will not be able to generate the I²C start condition. The MCU will not be able to transmit the I²C address.

Wondering why it worked for 3 months? Read on.

2) The lack of pullups is likely to damage any of those two ICs in my board?

Probably not. In this particular case (MCU, RTC, nothing else), definitely not.

3) Why was the MCU able to communicate with the I²C slave device in the first place? I²C requires pull-up resistors. But they weren't included in the schematic.

Probably, you have internal pull-ups enabled on the ATmega. From what I've read¹, ATmega have 20kΩ internal pull-ups, which can be enabled or disabled from the firmware. 20kΩ is way too weak for the I²C pull-up. But if the bus has a low capacitance (physically small) and communication is slow enough, then 20kΩ can still make the bus work. However, this is not a good reliable design, compared to using discrete pull-up resistors.

¹Not an ATmega guy myself.

update: In response I²C waveforms, which were added to the O.P.
The waveforms in the O.P. have a very long rise time constant. Here's what I²C waveforms usually look like

PIC18F4550, Vcc=+5V, 2.2kΩ pull ups. Waveform shows SCL. The rise time on SDA is about the same. The physical size of the bus is moderate: 2 slave devices, PCB length ≈100mm.