re: I am looking for a Business directory template which will have a search feature, alphabetical choices, and category search feature, pulldown
Any of these:
Choose the one that has appeals to you the most and show the data you want. Fox example Tile Plus shows very little.
re: Widget sidebar that will integrate with my theme’s widget by shortcode or sidebar widget.
There is the Widget Pack.
re: Also, do these templates integrate well with themes or are the totally separate?
template do have there own style which is one of the reasons they all look different. Other than the styles set to maintain their visual style, all other styles are inherited from the them so the do integrate well. That’s why neutral colors where selected for the defaults colors for template that have coloring.
Hope that helps!