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. As almost all routes have schedules which repeat on a weekly basis during each season, checking one week tells you much more than just that week. Actual season dates vary between airlines and regions, but even if you don’t know a given airlines season change dates you can certainly bring your search down to 1 week per month of the year and have a very clear idea of when an itinerary is possible.
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. Some, like the United page included in the question, fail to communicate that effectively or at all, others do so much more effectively. For example Hamburg airport’s destination map, where you do have to pick a date first, but once you’ve done so you can click the calendar to select a new date, and each day has a coloured icon indicating whether there are direct, connecting, or no flights to that destination.
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