While updating LineageOS, do I need to update Gapps too? I'm flashing a new nightly version of LineageOS. I'm just gonna flash it without wiping data/system since I already have it installed. However, I also use Gapps (stock variant). Do I need to flash an updated version of it too?
You normally don't need to re-flash GApps as most GApps installers will install an addon.d script which restores GApps automatically after flashing a new ROM that supports addon.d (LineageOS does so).

If you are using Open GApps (http://opengapps.org/) you are on the safe side.