First I'd make sure that mysqld is running. You can check that by typing
ps ax | grep mysql
You should get some output - if not then your mysql server isn't running.
If it is running, then you can restart it by editing the [mysqld] section of /etc/my.cnf (may be /etc/mysql/my.cnf or something else similar) and add the line:
skip-grant-tables
Then restart the mysql daemon:
/etc/init.d/mysqld restart
You should now be able to login without a username or password to perform maintenance. Make sure you remove the skip-grant-tables line from my.cnf when you are finished.
Login to mysql and to the database mysql. The following should do it:
mysql mysql
Now check your user table to see what it does look like:
SELECT * FROM user
You should get a few results back - each with a hostname (or %), a username, and an encrypted password.
You can then permissions and whatnot like you see on:
http://dev.mysql.com/doc/refman/5.0/en/default-privileges.html
As far as system mail is concerned - some research seems to say that new versions of ubuntu (Breezy) don't install the mailx package by default. You can install it yourself(and I would recommend it as it can give you useful information when stuff goes bad).
For dovecot, can you give the output of:
ls -ld /var/run/dovecot