Some champions (like Ashe) have toggleable abilities (for example her Q - Frost Shot). When I activiate such an ability, how does it proc an On-Hit passive like the Spellblade passive from the item Trinity Force? Does this behaviour apply to all champions with similar abilites, like Aatrox?
Toggle Abilities do not trigger an ability based effect
Now an Ability based effect is something like Sheen, Rageblade or The Maokai Passive which stacks up for every spell cast. This basically means that toggle abilities aren't considered spells.
You will however get stacks on Tear of Godess and the two items you can build out of it if you spend mana. Some good examples are Singed (Q costs mana every second activated) and Ashe (Costs mana every shot done with it).
Another thing that could be classified as a "Toggle" is the Ultimate of Nidalee. It won't give you stacks on a Tear like item but it will count as a spell cast (It doesn't cost any mana).
There are only two exceptions: The Toggle Abilities that do count as a full spell are Maokai Ultimate and Anivia Ultimate.