Biff, I will defer to Dennis, but I do believe you should repair areas that look like your top hat. See the line that looks like a tear in the grout? Water can get in there.
Your grout likely came in a white tube. It used to come in a little bag, thus the "cut off the corner" reference. The grout and liquid in the tube may have separated a little, so squish around a little to reincorporate.
Clean the white residue off the tiles and grout. I use Simple Green cleaner for stone surfaces. Wherever you see those rips in the grout, squeeze a little grout on your finger and rub it in in a circular motion. It will fill in the affected area. Don't worry about getting it on the tiles. Rub it in until you don't see the line anymore. Let dry completely and wipe the excess grout from the tiles with a wet cloth or paper towel. It comes right off.
I live in Texas, so the weather that my KK is exposed to is moderate. I believe the concern with any breaches to the grout is water getting in and freezing resulting in potential damage.