Is this multiple users on the same computer or the same user on multiple computers?
I would duplicate the user and on the same computer see if is repeatable, if it is I would then move to a working SAP GUI computer and log in there with the duplicated ID. I would also repeat this process with a working account that has no issues on both computer systems to see how much, if any, of this repeats.
For the logon problems I have seen the password once a day error if someone SQL savvy has performed some data manipulation on an ID to prevent the password from expiring.
For the other issues after logging in, I would think this is all controlled on the user's profile...