Personally, I would use the capacitive sensor to trigger a SWITCHCASE and have different brightness set to the LED strip. You need to figure out what you want the display to do... Immediately jump to the next value? Fade into it? Will it be doing anything without input? Are the numbers going to be preset or random?
You should use a high power transistor to switch the brightness for the strip (IE. TIP122).
I'm not very familiar with the MPR board, however the principle is the same in the sense that the input (The base of the art) should have a wire connected to it and it should go to one of the pins on the breakout board and that will be the sensor. https://learn.sparkfun.com/tutorials/mpr121-hookup-guide/capacitive-touch-sensor-breakout-board
I'm unsure of whether the board has an autocalibration for sensitivity as such a large sensor will be prone to feedback/false readings so that may take some debugging come time to run.
If you're going to be using an Arduino Uno, the regulator onboard should be okay for your input of 14v from the solar panel so you shouldn't need to fiddle with any regulators. However, for longevity, it may be prudent to stick a larger regulator to avoid prolonged high voltage, a +12v regulator can be had for a few dollars in a TO220 case.
As for the battery, you need to figure out how long you'd like the project to be running to determine the power required.
Will it be on all night long? Just the day time? Anywhere in between?