I downloaded SKSE (Skyrim Script Extender) and started Skyrim the normal way but I got popups that SKSE was not working properly. I then ran SKSE and it worked properly. But the question is do I really have to run this every time I play?
SKSE is required for some 3rd party mods to function. It will usually indicate on the mod's download page whether SKSE is required (at least on sites like Nexus.)
If you're happy to play Skyrim without your 3rd party mods, or with those mods potentially not working properly, you can run the standard executable and the game will work. SKSE will not be affected if you do this.
However, if you want to play with your 3rd party mods, you will need to continue running SKSE as your method of launching Skyrim.