Stopovers can range from a mere hour to even a full day (or more), so it's really up to you whether you would prefer paying less for your flights in exchange for doing some time at the connecting airport. The biggest issue is that time spent in airports eats into the "free" time you would otherwise enjoy at your final destination. If you do settle for a trip with a long layover, check into sights to see or activities to do near your layover airport. A 10-hour layover might provide the perfect opportunity to visit a famous landmark or explore a new city. To each, his own.
usually don’t charge you any type of fee so it never hurts
Nevertheless, it is entirely up to you whether you prefer to splash out on regular air tickets or put in the effort to score some of the amazing deals out there! But even if you're a very busy person and don't feel the savings are worth your time, you can still use a concierge service like those you get with good credit cards to do all the grueling work for you. You can never be too rich to save money.
You're not crazy for thinking that a flight price has changed after searching it a few times in your web browser. Based on the cookies in your browser, flight prices do increase when a particular route is repeatedly searched, as the site wants to scare you into booking the flight quickly before prices get even higher. Always search for flights in incognito or private browsing mode to see the lowest prices.