Without dates, how can I find out - efficiently and freely - when an airline's flying a route?

FlightConnections.com shows the days of the week on which a given flight is operated. I have found that their data is not always 100% accurate, but it would give you a better idea of which airlines operate a flight on which days.

For example, in the screenshot below, you can see that EgyptAir flies from Baghdad to Cairo every day except Mondays and Tuesdays, while Iraqi Airways flies the same route every day except Mondays and Wednesdays.

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

Generally speaking, airlines don't change their schedules very often. For most city pairs, at least one airline flies it as a daily route (or many times a day.) If you happen to discover a city that doesn't have daily service, you can follow this algorithm:

• identify an airline that flies the route
• for a week in your rough date range, discover which days the option is available (you may have more luck doing this on the airline's own web site rather than on an aggregator) by looking for a flight on each day of the week
• prepare a hypothesis such as "they appear to fly it Tuesdays and Thursdays."
• choose a nearby week and test your hypothesis. Refine if need be.
• optionally, choose a week 6 months away from the first and test the hypothesis again. This will let you know whether your proposed rule holds year-round or not

You can then also look through the "news" or "announcements" section of your chosen airline to see if they are announcing that their former once-a-week-service between X and Y is now happening twice a week, or that their "usual seasonal reductions" will be happening on a particular date. This should enable you to build up a reasonably confident pattern.

I used this approach to build a 26 leg trip that included a leg that happened only twice a week and another only once a week. The websites of the selected airlines were far more useful than larger search engines for this purpose.