You stated "Every 5-10 minutes, the 'internet' stops working. Except it doesn't, only website traffic stops." But then you stated "I cannot ping any website or IP." So, you are experiencing a problem that is not limited to just website traffic, which would be HTTP or HTTPS traffic. The ping utility does not use those protocols, instead it uses the Internet Control Message Protocol (ICMP). You state that you can continue using Skype and a BitTorrent client, but, perhaps, they too have lost network connectivity, but they just haven't yet notified you of a dropped connection, because they are still waiting on data from other systems. Do you see data continuing to be downloaded by the BitTorrent client or, instead, does downloading appear to have paused? Likewise with Skype, at the time the problem occurs are you receiving updates from the system at the other end of the connection?
Unfortunately, though I'm experiencing the same problem on a MacBook Pro running OS X El Capitan, I can't offer you a solution, only the workaround that I've used, because I haven't had time to determine the root cause in my case yet. I never have a problem when I'm at work, but the problem occurs intermittently at home, but only with my MacBook pro laptop. Someone else in my house isn't experiencing the problem with his laptop running Microsoft Windows, though we are both using the same wireless router for Internet connectivity. Nor do I experience the problem when using my wife's Windows laptop. What I've been doing when I experience the problem is clicking on the Wi-Fi icon at the top of the screen then clicking on "Turn Wi-Fi Off" and then "Turn Wi-Fi On". That restores the connectivity immediately, though, as you report, the WiFi signal appears to be good when the problem occurs. Yet I can't access websites and I can't ping systems. E.g.,
ping 188.8.131.52 fails until I turn Wi-Fi off and on. Likewise, if I'm participating in a teleconference using Skype for Business (Lync) where someone is presenting slides I don't see the current slide they are on until I turn Wi-Fi off and on and re-establish the session when the problem occurs.
Update: I believe I've identified the source of the problem, at least in my case. It is possible that the cause of the problem is the same for you. I believe I am losing Wi-Fi connectivity to the Internet because of a neighbor's router using a conflicting country code. Some wireless routers provide a country code in the information available to wireless clients in the vicinity. Apple systems running OS X may have problems if there are wireless routers nearby with differing country codes. In my case, someone apparently has an Arris router from Mexico, though I'm in the U.S. I believe that because another nearby router is using a U.S. country code, I'm losing Internet connectivity periodically. You can determine if that may be the cause of your problem by running the OS X wireless diagnostic program or by issuing the command
/System/Library/PrivateFrameworks/Apple80211.framework/Versions/Current/Resources/airport -s in a Terminal window. If you see differing country codes in the "CC" column in the output, then you may have the same problem. If it finds differing country codes, Apple's diagnostic utility will report "Wireless routers should only be used in the country they were originally obtained from. Failure to do so can result in performance and reliability issues for nearby wireless clients."