Does a MicroControllers resistance change with computation?
You can't really think about a microcontroller in terms of resistance.  It's built with transistors, and these do not behave like resistors.  

Generally, most microcontrollers (and most highly-integrated digital logic) are CMOS devices.  This means that all of the logic gates inside are built out of complementary pairs of N-channel and P-channel MOSFET transistors.  One characteristic of CMOS logic is that it consumes the most power when switching.  This is because the transistor gates look like capacitors, and these capacitors have to be charged up whenever the logic level is changed.  At idle, little power is drawn.  The power consumed during switching is called dynamic power.  The general formula for CMOS dynamic power consumption is P = C * V^2 * f.  So the power consumed is proportional to the capacitance of the traces and transistor gates, the frequency, and the square of the voltage.  Now, this is not the complete story - there will also be some leakage current that makes its way through the transistors.  The leakage current increases with higher supply voltage, and it increases as the transistors get shrunk down on newer processes.  

Any component inside of the chip that switches will consume power.  So yes, any peripherals that are enabled will increase power consumption.  Different processor instructions will consume different amounts of power as they will activate different logic.  Even the data being processed can have an effect on power consumption.  This is why it is possible in some cases to recover cryptographic keys by looking at the power consumption during computation.