No, but very unethical
It's not illegal, but it's his private mail; What if he goes throw your things, would you like it?
It's not illegal and if you have reason to doubt then maybe it would put your mind at rest?? However, like mentioned above it is private and you must be careful not to read too much into something if you do find it. You could use people search websites which only bring back public online info about your partner but will tell you whether he's joined any forums you didn't know about, whether there's any pictures of him with this other person. Try www.yasni.com