Do various soaps work better when dissolved/used in different water temperatures?
To start, soaps and detergents have two different chemical makeups, meaning that they act different under different conditions.

For general soaps, here is an explanation of how it works:
http://chemistry.about.com/od/cleanerchemistry/a/how-soap-cleans.htm

As a general chemistry note, as the temperature of a solution increases, the ease of a chemical to change (whether it is to react or change shape) increases.  This means that colder temperatures will allow a chemical to remain in the same state.  If the soap depends on a specific conformation/shape to act then colder temperatures may be preferable.  However, if a reaction needs to occur in order for the soap to act some heat may be required.

For Tide Pods, it states that it "dissolves quickly in hot &amp; cold."
http://www.tide.com/en-US/product/tide-laundry-detergent-pacs.jspx

The chemical makeup of Tide Pods is composed primarily of salt compounds and enzymes.  Both salts and enzymes tend to operate better at higher temperatures.  So as long as you don't pass the denaturing temperature of the enzyme, higher temperatures would be preferred.