If I purchased a brand new utility trailer with all of the appropriate safety features on it, do I need to bring it with me to the DMV when registering it and getting my tags/plates?

Hey Bob it depends on the length so it's best if you call them they'll guide you accordingly

I looked through laws and regulations pretty thoroughly, I would say no you don't need to bring it just the relevant paperwork. Title, bill of sale, etc. They may require you to go to a shop with the trailer for an inspection. I'm unsure of Oregon's policy on inspections of trailers.