Do I go to the Witcher 2 site and manually install patches? Is there a patch page on GOG I can't find? OR do I just redownloaded the whole updated game from dog and install it over my old Witcher 2 install? (or do I uninstall my current version first then install the new version)?

Also I am halfway through chapter 2. Do saves ever get broken when updating the game?

I believe the GOG version is fully patched already

There used to be a bunch of pages for patches, but they don't seem to work anymore. This page seems to have replaced them: http://en.cdprojektred.com/download/enhanced-edition-patch-3-0-separate-files/

Looking at that, it might just be easier to download the whole thing again if your current game version is anything below 3.0 (at which point it became the "Enhanced Edition"), honestly. I remember the EE update being rife with problems for those who were updating instead of downloading the whole thing, though some of us were lucky enough to get through it without any problems. Yes and no. If you're playing version 1.0 (as in, a physical disc version that hasn't been updated) or 1.1, those saves are incompatible with later versions of the game, but if your current version is 1.2 or anything beyond that, patching won't break your saves.