arduinopower supply
I have a 5V Arduino Pro Mini that I want to connect with this 3.3V nRF24L01+ module. (On the other end I have the same nRF24L01+ module connected to an Arduino Uno which has a 3.3V pin that I know is safe to use.)
The nRF24L01+ modules I have are these.
The Arduino board this question relates to is this one.
I think you're killing the tantalum capacitor across the input. In fact, it looks like it's bulged up in your photo.
Although the capacitor may be rated at 16V (I don't actually see a rating anywhere, but I'm inferring it from the size and input voltage spec) seasoned engineers will never use tantalum capacitors in this sort of situation. The input current needs to be limited by several ohms and/or the rated voltage should be something like three times the operating voltage. Your wall wart can probably supply several amperes.
Years ago, this information was anecdotal, but today you can find it buried deep within the manuals for such parts. They bury it, because it means you very seldom should use tantalum capacitors, and almost never across power supply rails unless the current is limited to much less than an ampere.
Many have been burned by this (perhaps literally). Often failures take place long after product has been shipped, and can cause localized charring, meaning (if your standards are high, as in aerospace or medical) the entire board must be scrapped.
The ATmega168 has separate pins for I2C (PC4:5), SPI (PB2:5), and UART (PD0:1). There will be no conflict between the devices provided you use the MFRC522 via either SPI or I2C. Note that the Arduino Pro Mini is programmed via its UART, so you may need to disconnect the HC-06 when programming via the serial port.
Best Answer
You could use a 3.3V voltage regulator. See DigiKey parts here. It looks like all of them accept a 5V input and I set a 20mA output minimum (the max supply consumption of your module is 13.5mA)