Question 1: Most (all? I don't think I'm aware of any which don't) major travel search engines allow you to search with flexible dates, including flights within (for example) +/- 3 days of the date you selected. You can also typically specify that you only want direct flights.
If you're interested in destinations available to/from a specific airport or via a specific airline, the Wikipedia pages of most major airports and airlines contain lists of the destinations they serve. As of course do their official websites. How effectively any of these sources communicate frequency of a given route is variable.
I'm not answering question 2, as
a) travel.SE is not your travel agent, and
b) searching for flights as above inherently reveals this information to you anyway