MySQL connection doesn't work anymore
Here is one thing I quickly see is missing : Each [mysqld] needs to have its own datadir. No two mysqld groups can share the same datadir. You need to allocate different datadirs in different folders and make sure you run "chown -R mysql:mysql (datadir folder)" on all datadirs.

Your /etc/my.cnf should look something like this:

[mysqld_multi]
mysqld     = /usr/bin/mysqld_safe
mysqladmin = /usr/bin/mysqladmin
user       = root
password   = ***

[mysqld]
port       = 3306
datadir  = /var/lib/mysql1
socket   = /var/run/mysql/mysql.sock

[mysqld2]
port       = 3307
datadir  = /var/lib/mysql2
socket   = /var/run/mysql/mysql2.sock


Check out this link from the Pythian Group on how to start and stop the individual DB services.

From my answer, you should see that there must be a complete copy of the data in /var/lib/mysql1, /var/lib/mysql2, etc.