But, I would chissle out the grout and being careful not to damage any tile, then re grout.
If you feel that you might want to experiment and grout only the affected areas, you might try a small part.
If the grout is old and stained, matching will be a big problem.
No, you do not need to remove the tile unless the break lines have caused some tile to loose its grip.
After this is done, be sure to seal the grout with a silicon sealer.
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