The arduino interface won't work as-is, but the rest looks good.
When the switch is on, the arduino's transistor will probably not like the emitter and collector both connected to +24v while the base is at +5v or whatever the arduino gives you. That's -19v, which is probably outside of the spec.
When the switch is off, the arduino will only be able to supply about +4.3v to the load (5v - Vbe) because the transistor is then operating as an emitter-follower.
The concept should work, but you'll need a slightly more complex interface from the arduino. Probably use either a PNP or a P-channel FET across the switch with a pullup resistor to +24v, then pull that down with either an NPN or an N-channel FET that is driven by the arduino.
As for your other questions, go to mouser.com or digikey.com or any other electronics warehouse and plug your requirements into their parametric search engine, one parameter at a time, most important first, and select as many options within each parameter as will technically work for you. When you run out of requirements, sort by price at your required quantity or round up to the next common price break.
Sometimes, the specs that you care about are not listed in the parametric search. In that case, specify what's there, sort by price, and start reading datasheets. For example, the base resistor for your ground connection depends on the maximum base current and the transistor's gain. You want enough base current to saturate the transistor (make more current available to the load than it will use), but don't exceed the maximum.
The big question is what exactly it means for the chuck key to be in the holder, and how that is sensed. Once you have a electrical signal of some sort, it should be a simple circuit to use it to drive a relay.
A nomally-open (NO) relay sounds like the right thing here. The relay has to be energized as a result of the sensor indicating the chuck key is in the holder for the unit to turn on.
The circuit could be as simple as a low side NPN transistor, emitter to ground, series base resistor, and relay between collector and the power supply. 24 V is a common relay voltage, so finding a relay won't be a problem. Make sure to add a diode across the relay coil in reverse polarity to give the inductive current a place to go when the transistor shuts off.
The above works when the detector signal is high when you want the device enabled. If the other way around, then a high side PNP switch works. It's basically a mirror image of the low side NPN switch circuit.
We can get into details once you have some real specs on the detector.
Best Answer
In this case your best bet would be to use digital POT, something like MCP4141 or AD7376. This one chip alone will replace all the resistors and switches and is extremely easy to control by MCU, with probably the smallest footprint possible.
For non-MCU control there are hundreds of miniature rotary switches available dirt cheap.