Two-part question: what are some things you do that require cold water, and other things you do that require hot water?

Oh that one's easy for me.  I work on muscles all day long.  Heat makes the muscles relax and cold makes the muscles contract.  I use hot towels and Bio-Freeze on tight, constricted muscle tissue.  When the tension and "knots" subside, I add cold compresses.    

I do this all day long.  I also use hot stones and cold stones.  And I use hot and cold water to get them to that state.

Cold: When making nitroglycerine, the nitric acid-glycerin solution should be placed in a cold water bath before adding the sulfuric acid.
When making tempura batter.
Hot water: Making tea. When I shower.

I use cold water to make ice, hot water doesn't work.

I've used boiling water to exterminate bed bugs from my furniture. 

not sure

Cold water....laundry
Hot water...dishes, cooking, my bath

Treatment of a sprained ankle I sustained in college called for both - alternately immersing it in hot water and ice water.