How should I protect wood gates in a damp climate?
"tends to rain" :) Wood will shrink and swell. You can't stop it. Rather, allow for it. You can use wood naturally resistant to decay. Not sure what's available in your locale, examples include redwood, cedar, teak, cypress. You can also use pressure treated lumber, in which case, be sure to apply preservative to cut ends. Field applied preservative will help for a while, but it will not last. You should avoid paint, the shrinking and swelling will quickly cause failure. If you must paint, all surfaces of every piece should at least be primed, if not fully painted.

Use corrosion resistant hardware and fasteners. Stainless steel if possible. At least hot dipped galvanized. Not the much thinner electro-galvanized. For hardware, a durable factory applied finish such as powder coating would be OK. Don't waste time with fine joinery, big and clunky is more appropriate. This is not an excuse to be sloppy, the big and clunky joints must be well made. The frame's primary diagonal should be low at the hinge side so that it's joints are in compression.