There's a clear need for a safety device like yours. Good luck with the project. Will you be able to compete price-wise with existing child safety devices such as this ?
As Bob says. However, I got f=1.5/RC by running a simulation with LTspice and measuring the frequency.
The book figure was presumably for TTL logic, not CMOS logic. Frequency is a function of the hysteresis, which is quite different for the two logic technologies.
For a 40106 the frequency will be ~ 1.5/RC if the IC is lightly loaded (i.e. has a feedback resistor which draws < about 1mA).
Bear in mind that datasheet values are "typical". Because of manufacturing tolerances no two ICs will have identical properties. The R and C values also have...
If the pots are part of a Baxandall tone control network I think linear would be needed; otherwise you'd get a rather lop-sided effect with the boost range being much greater than the cut range (or vice versa).
The schematic editor in LTspice (free download from Linear Technology) is easy to use. You can also simulate the circuit operation (though learning how to do that well is a bit daunting).
You would use the logic Fault = "contact energised AND coil de-energised" OR "contact de-energised AND coil energised". Probably a couple of voltage-dividers and an XOR gate would suffice for that.
If you want just to detect coil failure you could use a Hall sensor to measure the coil current, then a window comparator would let you know if the current was too low (coil open-circuit) or too high (shorted turn). Alternatively you might be able to arrange that actuation of the solenoid would...
Please define 'best'. Cheapest, quickest, most reliable, easiest to construct, minimum components, .........?
Personally, I would discard the relays and use logic circuits and power MOSFETs to drive the solenoids.
As you don't specify all the components there is no obvious solution. It seems each component would need its own failure-sensor. The type of sensor would depend on the type of component. The sensors could operate respective LEDs. It doesn't sound cheap or simple. One name for it is 'failure...