@SVXX, I agree with the intermediates in your second mechanism. Obviously, the final product is wrong as you removed a carbon. (Your use of curved arrows is also errant. In my class, I would require you to maintain the C-Mg bond as no curved arrows show a change in those electrons nor can you start a curved arrow with an atom, especially Mg2+.)
@johnnyd, if the halogens are geminal or vicinal, then you cannot form the Grignard. With longer chains, then you can.
If you deprotonate a diol with a Grignard reagent, with small diols, I expect you would get the chelate as suggested. If the chains are long, and concentrations high, then intermolecular salts are likely to be prevalent. I would use rates of ring formation to be a guide for making this type of prediction.