5W20 sounds wrong, 5W30 sounds right, but I have a 2000 CR-V, so you would know. The oil that's in there will be fine.
I use 10W30 for winter and 10W40 for summer, so the 5W will be fine.
Oil brand preference is a can of worms I don't want to open, but I use Walmart dino oil changed as close to 3,000 as possible and I have never had an engine related problem on any of my cars (BMW and Honda).