Is it permissible to kill mice, and is it permissible to leave them in the sun until they die inside the trap, and is it permissible to put the mouse in water until it dies, out of fear of them?

It is not permissible to kill mice by methods that involve torturing them, such as leaving them in the sun until they die or putting them in water until they die, as long as there are other methods of killing them without torture. This is based on the saying of the Prophet, peace and blessings be upon him: "Indeed, Allah has prescribed excellence in everything. So, if you kill, then kill well; and if you slaughter, then slaughter well. Let one of you sharpen his blade and relieve his animal."