Where did that come from?

The custom of a man tipping his hat to a woman was derived from a custom of knights in  full armor who would raise their awnings  when they met each other on the road to identify if the other was a friend or an enemy. When hats became a part of a man's daily attire, tipping his hat was the evolution from raising the awning.

The custom of a man walking on the outside of the sidewalk and his wife/female companion walking on the inside of the sidewalk was due to the fact in European towns before indoor plumbing and a proper sewage system was available, people dumped their trash and waste water in the gutter and a woman's clothes could get splashed so this was the man's way of protecting her clothes from ruin.