One point at a time.
would be used to turn the device off without resetting the 7-segment display
If the 4026 loses power, the count is gone - period. You could enable the blanking input to extinguish the 7 segment display though.
three switches: one to reset, one to turn off/on the device and one to add to the count
The reset is straightforward. Bring the reset pin high and the count goes to zero. No need for switch debouncing.
On and off the entire device? As earlier stated, if you wish to store the count, the 4026 will require power.
Adding one to the count will require a debounced switch (just like it sounds - mechanical switch contacts will physically bounce before coming to rest. The counter is fast enough to see this bounce as a series of valid pulses) - look up switch debouncing. If you need to manually insert a count "into" an existing clock line, then you can do so with small signal diodes.
possibly a piezo buzzer to read the count on the 7-segment display (?)
Don't understand. Do you mean a single beep every time the count is incremented, or (say) - at the push of a button have the buzzer beep 7 times if the display is 7?
Assuming the former. although integrated piezo buzzers do exist, the bare disk type will require an oscillator.
To just chirp, a piezo "buzzer" (with integrated oscillator) would require at least a transistor (so as not to load the clock signal) - a sustained beep would require a time delay, like a "one shot".
If you just have a bare disk, a 556 dual timer could do the buffering, one shot, and oscillator duty.
but did not get very far.
How far? Noting that the purpose of and connections to, the 555 on your diagram is all forked up, as well as why are you pulling the output low,
What is it supposed to be doing? If continuously counting, then it has to be wired as an "astable multivibrator". If just debouncing the switch, "non resettable one shot".