The problem with a circuit that fits all is that it is optimal in none. So, in order to support the wide voltage range that you need, you'll need to pick a part that might not be ideal in some aspects such as size, output ripple, etc.
THere is no "configuring" the input voltage. The input is a voltage range, and in your case you want 20V as max, 5V as min. However, your requirement to go from 6-20V and others means you're looking for a buck/boost solution. The output is usually easily configurable with a pair of resistors in the feedback network.
The biggest problem with your specs is that you didn't specify output current. This is one of the most important parameters and choosing a buck/boost device (or any power supply device) is impossible without knowing this parameter.
I looked online and a nice part that will meet your specs (as much as I have) is LTC3115 from Linear, but this part is not cheap.
Your desire to reduce complexity with one solution is understandable, but you will be better served by finding parts tailored fit for your solution, especially as far as reducing costs. Most DC/DC converters are based on the same 3 topologies and are not very difficult to use.
Yes. The current through the converter is determined by the load. I doubt the monitor is pulling all 5 amps from the original supply anyway. Just throw an ammeter on the ground of the pc side and measure it to be in the safe side, but it should work without modification. With 100% efficiency the input would be, at max draw, 12v 10A, for 24v 5A. In free air inside the pc case, with fans you should be alright assuming 4A load draw
In simple terms it's called a boost converter and, fairly efficiently takes a lower supply voltage and transforms it to a higher supply voltage. You can buy plenty of off-the-shelf items that will do the job - try ebay.
One word of warning though - if it's to be used for audio applications (such as the guitar pedal) you may need to boost up to (say) 18V then use a linear voltage regulator to generate 15V. Typical candidate would be an LM7815 for this.