Would updating to OS Sierra remove my third party apps?
To add to @Tetsujin 's answer as well as give it more depth, that entirely depends on the application it is and the version you currently have. Whatever version of MacOS was out when your last update was released is the version that update was made for, and the changes to MacOS from that point onward could all harm app stability or break the app entirely... or do nothing at all.

Different apps have different needs, and when a part of macOS gets changed and that app needs it, the app sometimes needs updating to work consistently again. Depending on the apps needs this is more frequent on some than others. The built in Stickys app for example, or TextEdit I doubt ever need updates ever because of their simplicity. The updates that come out for them would be new functionality or eye candy.

On the other hand some applications are very picky in their function. Take the older version of ProTools HD as an example, which needed an $8000 processing card in your Mac Pro to run. Updates were very frequent and the app would need several updates per MacOS release, but it pushed the hardware to the limits.