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You probably mean something like this.
Usually they come with software that allows you to "program" it, by associating a keypress or sequence of keypresses to each key. Once it's programmed, you can usually move it from one computer to another and it will still work, but some models might need you to install a driver.
Another way to solve your problem doesn't necessitate buying a separate keyboard. Depending on your OS, you can install a different keyboard layout that will allow you to type accented characters with ease. For instance, Linux allows you to define a "Compose" key. Then you just type Compose then ^ then e to get ê, and so on. You can also define other maps such as Compose, +, then - to get ±. There are a lot of predefined sequences for characters, such as Compose c / which gives ¢. It's very useful.
True. All depends on how much your time is worth, and how much you're willing to spend.
Some switches, a Teensy, diodes, wires, and a plate to hold them would technically work for the cheap, plus several days worth of your time. A relatively affordable keypad from ebay would work, if you don't mind figuring out how to program it. Or, a programmable keypad is basically plug'n'go, albeit more expensive.
Only thing i know of is this but you have to assign the keys to do what you want
Genovation ControlPad CP24 USB HID https://www.amazon.com/dp/B00O1ASXKO/ref=cm_sw_r_cp_apa_QULTBbGZR16B5
This is pretty neat. I was actually looking to get one of these to do something like this.
Some of them sends numpad codes, but I'm not sure if there's any that's programmable that is readily available and affordable. If you don't want to use a software solution for macros, you may want to consider building your own instead.
I did a quick search and found a few :
http://cherrycorp.com/product/g84-4700-programmable-keypad/
http://www.fentek-ind.com/xks16.htm#kp24xkusb (rubber dome)
you still can, more often than not they are small keypads rather than fully programmable 101 key keyboard. or keyboards with 6 or 8 macro keys on the side.
www.amazon.com/Genovation-Controlpad-CP24-Keypad-CP24-USBHID/dp/B00O1ASXKO/