MIPS invert - why are the 2 operands always inverted first thing?
I don't understand why is the inverter at the begining.


That is useful for subtraction. You can calculate A-B by calculating A+(~B)+1 - i.e. invert B, and turn on the carry input to add 1.


  Another thing is that part that has a 1 on it (in the picture right before the inverter or as part of it) - what's its job?


That is an unconventional symbol for an inverter, I think. The box with a "1" is a buffer (it doesn't change the signal) and the dot on the output inverts it.


  why are there 0 and 1 on the bitinvert?


Those are muxes (short for multiplexers). If Ainvert is 0, then the A1 signal is passed to the output. If Ainvert is 1, then the inverted A1 signal is passed to the output.


  What do the 0,1 and 2 in the ALUMUX mean? Does the ALUMUX decide which of the results gets passed forward according to the current instructions?


Same thing as the Ainvert and Binvert muxes, but now there are 3 choices instead of just 2. Yes it does.