Bolded emphasis are my answers.
-May the explorer, for instance, use the temporary weapon level increase from Surprising Discovery (green adventure card) when another character is fighting a beast (another character is the resolving character) and he is not in the fight? (not supporting and not resolving)
No. The Surprising Discovery weapon is put into the Future resources area and is not available for the current actions
-Also, may he use it in a hunt where he is supporting? (but not the resolving character)
No. Since the Surprising Discovery weapon is in the Future resources you do not have access to use that weapon yet.
-May he use the increase when resolving the red die (Hungry Animals) during weather effects and he is in the camp?
Yes. This is because the Future Resources are now gathered for camp after all the actions have taken place.
-May he use the increase when he is spending the night outside the camp, but the camp has to fight a Hungry Animal during the weather effects?
No. When a character spends the night outside of camp any resources or tools they gathered/made are placed on their character board instead of the future resources area. The temporary weapon from the Surprising Discovery must remain with that character and cannot be used for the camp weather.
However, the character may use any tools, weapons, food, etc. for themselves.