Or should I use 7555s, which draw less current, and leave them powered on? Are they significantly more expensive? How long could I expect the button cells to last with a 555 vs 7555?
The 7555 from Analog or Maxim, or TI's TLC555 or LMC555 draw significantly less current while idle than the legacy 555 ICs like LM555. The LM555 has ~5mA standby current at 5V, 16mA @ 15V. The newer ones are in the microamp range, like 250 µA (0.25mA) for the TLC555, or 50 µA for the LMC555. Since the standby current difference is an 1 or 2 order of magnitude lower, they will last 10 to 100 times longer.
The tradeoff is that the newer 555s have a smaller voltage range, 10V max for the ALD7555. Then again, they also work at lower voltages, some down to 1.5V like the LMC555 instead of the 4.5V of the LM555. There are more differences that may be important, but not really.
The cost difference is small. ~25 cents at 1k units, instead of 7 cents.
Based on the schematics and end goal, even CMOS 555 timers won't work well for your application. Coin cells don't have much energy. The common cr2032 provides 250mAh at low draws. Just the 555 timers would drain that in 1000 hours, not including active power draw. Add a AND IC, a Flip Flop, the Buzzer and Motor and Transistor? Under a month, max.
Solution? A microcontroller to replace all the logic. A single microcontroller can handle keep the time (5 second counts up to 1 minute), no need for the flip-flop or the AND ic, or inverter. Everything in code. In deep sleep like the MSP430 can have just 10µA current. It sleeps in Low Power Mode 4 or 4.5, wakes up when the push button is pressed, turns on its clocks, not even only 100µA.
The biggest problem is simply how much current is used by the buzzer and motor. If you power the motor and buzzer off AAA or AAA batteries, and the microcontroller on a coin cell, you could get a year. Assuming you keep the microcontroller current under 30µA average, then it will last 1 year. If you just power everything off 2x AAA (3V 1500mAh) or 2x AA (3V 2800Ah) you will get years (depending in how often it is triggered).