If you buy an item at an auction and you win it on ladder, you will have the item for the rest of the ladder. When it ends, every character and their items get transferred to non-ladder. So yes you keep it on non-ladder when the season ends. Since ladder/non-ladder characters don’t play together, trade can’t be made from ladder to non-ladder
And you can’t transfer ladder characters/items to non-ladder in the middle of a season as far as I am aware of.