What is the mechanism of the reaction of phenylmagnesium bromide and CO2, including the work-up steps?.?

You have phenylmagnesium bromide, Ph --MgBr.   The Ph-Mg bond is polar, with more electron density around the Phenyl carbon and less around the magnesium.  The electrons in the bond attack the carbonyl of CO2.   One pair of electrons leaves one of the double bonds and goes onto one of the oxygens.  A lone pair on that same oxygen now has to move and reforms the double bond between the oxygen and carbon.  In order to compensate, a pair of electrons in the 2nd carbon-oxygen double bond moves up onto the second oxygen.  Now you have

Ph--C=O -O-   This second O is attached to the carbon still, you just can't draw it on here.   The MgBr+ is ionically hanging around this negative oxygen.  Once acidic water is introduced, the negative charge on the oxygen takes a proton .  now you have a carboxylic acid (COOH) on the phenyl ring.  PhCOOH.

I purposely did not draw the mechanism out so you can work in out on paper yourself and understand the mechanism better.