For recognized events, you don’t need USEF membership until you’re planning to show Preliminary (which you have to qualify for with satisfactory results at Training and/or Modified level). I believe you can enter Starter level at recognized events without being a USEA member, but anything higher than that DOES require USEA membership.
Re: horse registration for USEA, similar rules to the rider. Horses at Prelim & above need Full registration status, but those competing Beginner Novice through Modified can have Limited registration status. I’m not positive whether Limited registration status has any effect on qualifying for awards/championships and the like.
For dressage, the rules are different and as mentioned above, can be more driven by the organization hosting the show. As a general rule, you’ll need USEF membership for USDF rated shows, but if you’re not planning on doing many dressage shows, you may be better off paying the non-member fee.
The USDF has a nice table on their website that outlines the differences between Group Membership (what you’d have as a member of the Oregon Dressage Society or any other GMO) and Participating Membership (what you have if you join directly via USDF). Again, if you’re only planning to do a handful of shows, Group Membership may be sufficient, but if you’re planning to campaign/want to qualify for awards/championships, you’ll likely need the Participating Membership. (And yes, you can have both!) Again, if a GMO hosts recognized shows, they may require you be member of their organization or pay a non-member fee.