The soap is a surfactant to make the vinegar stick to the weed leaves for longer.

It's not necessary, but it probably lets you use a little less vinegar, so it's probably worthwhile. I don't add soap, I just spray straight 30% (agricultural) vinegar in the small set of areas where a torch would be dangerous.

Dried vinegar does not irritate dogs. They will avoid the area while it smells like pickles.

A better chemist than I will hopefully corroborate this, but I think that the strength of smell is directly correlated to the reactivity of the acid. So when the smell is mild (i.e. near the level of household vinegar (5%)), the risk to skin and mucous membranes is low-to-zero.