1) Well, the "phase regulator" is actually technically not labeled in your picture.  The PWM controller chip is technically the phase regulator which is the chip with the pink dot on it above the #6. But, don't worry about it because that chip doesn't get hot.
I assume you must mean "MOSFET" when you say "phase regulator".  The MOSFETs are #4.
 
2) #4: definitely.  Of the items you have labeled, #4 is all that really matters.
#5: meh, might as well (if it is even an option).  #1: meh, might as well (if it is even an option).  The others, no problem.
 
3) Are you serious?  Just look at where the thermal pads were on the cooling plate you removed.  Everywhere which was cooled before should still be cooled now.  In other words, cool the row of MOSFETs labeled as #4 in your picture, the row of three MOSFETs over by the pink-dotted PWM controller chip (not labeled in your picture), and the memory (not labeled in your picture).  Any more than that is optional.