1) How to check the status of DB2: It depends on what you mean. You can check the instance using db2gcf. You can check the databases of the instance by the command: "db2 list active databases".
2) There are 2 kinds of logs in DB2. There are the transaction logs DB2 uses to make sure that the database is always in a consistent state. These are not meant to be looked at by the user. There is the diagnostic log file that the instance uses to record things into. This is the one you probably interested in. There is one for each instance. It is usually located under the instance owner's home directory in sqllib/db2dump/db2diag.log.
3) Just about everything in DB2 can be monitored. What you need to look at depends on your particular situation.
4) There are a lot of things that can be looked at to check the performance.
I suggest that you get on the IBM site for V9.1 and read through the documentation there to better familiarize yourself with the product.