I want to update my Copay wallet but not sure if I'm supposed to install the new (updated) version of the wallet or what? Will it not mess with the old version or do I have to remove it first? Or do I transfer my btc afterwards?
You can just remove the current version and install the newer version since they are storing your details on your pc, not in the application itself.
What you can do as an extra action, is storing your backup phrase somewhere.