Well different conditions work differently...
Items and recoil damage are taken into effect after a move has been used. Therefore, if a Pokemon with a Life Orb KO'd a Pokemon, the Pokemon will faint first, then the Life Orb will deduct damage. However, if a Life Orb Pokemon delivers a final blow to the last opposing Pokemon, therefore you have won the battle, so the damage of the Life Orb wouldn't be taken into account. This works exactly the same for recoil damage - since both the Life Orb and recoil damage are aftermath damage.
For weather conditions, the Pokemon's speed determines the order of which Pokemon are knocked out. Lets say 2 Pokemon (yours and the opponent) are left with 1HP and a previous sandstorm is still in play. The fastest Pokemon will be knocked out first; so this may be an advantage in concern to slower Pokemon. This works the same for 2v2 and 3v3 battles. If more than one Pokemon have the same speed, a Pokemon will be chosen randomly to faint first - similarly to how Pokemon move in battle.
To verify this, you can try on Pokemon Showdown, or watch various online videos to check the order of Pokemon fainting.
Hope I helped! :)