s2Member will log the IP when a user registers as a part of their profile. s2Member also records how many times a user has logged in as well as the timestamp of the last login. I believe all of these values are just the last login incident and not a rolling history that you can view. These options are shown as columns in the user list. If you don’t see them, you’ll need to activate them in the “screen options” dropdown.
If you need to maintain a more thorough log history of all users’ logins/logouts, then you can use a separate plugin to accomplish that.
That’s strange. I use the Simple Login Log plugin, and it records the activity of everyone who attempts to login.
In fact, since one of the reasons for that plugin’s existence is to track failed logins, the behavior you are reporting is not how it should be working. At the point of attempting to login, the plugin can’t know whether that person is a member or not, let alone what role they have.
The behavior you are experiencing therefore suggests that there is some other problem on your site — most likely a plugin conflict somewhere.