Electronic – arduino – How to get an Arduino (5 V) and MPR121 (3.3 V) to talk

I would recommend reading the datasheet again for that device. It sounds like CTRL0 and CTRL1 can also be used as output enables depending on the devices control registers MUX and BUS. This would explain why when you assert CTRL0/CTRL1 the oscillator output is disabled. Another thing to note is that the datasheet says the register values are store on an internal EEPROM, but only after a write command is executed. It could be possible that you think the registers are set one way, but when you power cycle the board the previous values are being loaded from EEPROM. I've worked a lot with I2C clock chips and every time things weren't working it was because I miss interpreted one of the register values. Write each register out on initialization then read them all back to make sure you initialized them properly. Compare the results to the datasheet to see if they make sense.

EDIT: It appears that the CTRL pins can also be used to place the device in a power-down mode which would explain why it's not responding to I2C at times.

On how capacitative touch "works", have you had a chance to read through the CapSense library page on the Arduino Playground? It's a decent overview. The technique is typical - measure how long it takes to "charge" a metal plate up to some voltage through a known resistance and infer the capacitance from that time interval. There's also a very instructive lab exercise at Cornell's microcontrollers course website.

What are the physics going on here? When you bring your hand closer to the plate you are increasing capacitance (by narrowing the distance between "the plates") - see also wikipedia entry on capacitance. You kind of serve as the plate of the capacitor connected to GND. So yes in principle driving an actual plate to GND through a transistor could work. Furthermore, if your Arduino is plugged into the wall, you most likely don't need to worry about whether you're at "the same" GND, it will be close enough. You might have a slight issue if you're Arduino is battery or USB powered though.

As for the actual circuit. I would actually try just connecting an actual digital pin to the plate. When you want to simulate "no touch", turn the pin to an input (without internal pullups enabled). When you want to simulate "touch" turn the pin to an output set to LOW. The only reason for using a transistor is if you have a current demand that exceeds the rating of the pin (about 40mA I believe), and this case really doesn't qualify.