How does shyvana's q proc on-hit item effects
Solution 1:
Most of them, including your own list, will proc twice. And when in dragon form, they will even proc twice on each target hit by Twin Bite.
Notable exceptions are on-hit effects with an internal cooldown, such as Spellblade (e.g. Sheen, Lich Bane).
See: http://leagueoflegends.wikia.com/wiki/Shyvana/Ability_Details