Chips may operate from 2V to 6V or whatever, but they will behave differently. At lower voltages the speed goes down and the drive capability may drop dramatically. 2.0V is an appropriate cutoff voltage for a CR2032.
4000 series CMOS logic is only specified down to 4.5V. 74HC and some newer families work much better at lower voltages. Some newer families of logic may be marginal at the voltage of a fresh CR2032 (too high), others are specified from 0.8 to 3.3V so are ideal.
CR2032 batteries are specified at modest current draw (200uA) and the voltage starts to drop significantly above 1mA or so, so keep the average draw low. If you have sharp current spikes you should add a lot of capacitance. The voltage drop due to internal resistance also increases greatly at low temperatures. You will only get about half the nominal capacity at -10°C and 2mA.
The 555 type circuit timing changes if the supply voltage is not stable within the timing cycle since the charge and discharge thresholds are 2/3 and 1/3 the supply voltage. For example, if you turn an LED on when the output goes high the supply voltage will drop and it will take longer to discharge the timing capacitor than with a stable supply. This can lead to changes in output timing that vary during the battery discharge or with temperature as the internal resistance changes.
Most things you can do with a bit of logic and a 555 can be done better and possibly cheaper with a small microcontroller.