Should I pre-cool my cooler before packing it?

Yes, do it. An hour before you pack, toss a bag of cheap ice into the empty cooler, close the lid, and walk away. This chills the walls so your real ice doesn’t have to do that work later. Dump the pre-cool ice after 30–60 minutes, then load your food and fresh ice. For a long trip or hot weather, it’s worth the minimal effort.