I would imagine you would want an under-deck gutter system for a usable patio. Basically consists of a ceiling that slopes to a gutter and then carries the water away.

If the framing is not treated, don't know what to tell ya. There's a whole host of detailing methods required for water with a deck even beyond the treated issue.

I live in a similar climate and building a covered trex deck over a patio and my builder told me we should put a coat of something on the beams. I want it to look nice underneath on the patio since we will be using it a lot but we don’t need water management since the deck is covered. The underside of trex transcend isn’t the most beautiful. We are trying to choose a color for the joists. Looking at sherwin Williams super deck in solids-caribou looks like it might be a close match?

Stain it something darker or it will show a lot of streaks over time.