Is using ASM a pre-requisite for RAC

DISCLAIMER : Not an Oracle DBA

Use srvctl, the grid control program that allows you to manipulate parts of the RAC Cluster

Usage: srvctl <command> <object> [<options>]
    commands: enable|disable|start|stop|relocate|status|add|remove|modify|getenv|setenv|unsetenv|config
    objects: database|instance|service|nodeapps|vip|asm|diskgroup|listener|srvpool|server|scan|scan_listener|oc4j|home|filesystem|gns
For detailed help on each command and object and its options use:
  srvctl <command> -h or
  srvctl <command> <object> -h

Examples

srvctl stop database -d mydb 
srvctl stop instance -i mydb1

Please get an expert to perform these, especially since you want to do something with ASM. If I recall, ASM is a proprietary Oracle file system. Using srvctl would be better than trying to handle it yourself.

Here is more help on srvctl

$ srvctl stop database -h

Stops the database.

Usage: srvctl stop database -d <db_unique_name> [-o <stop_options>] [-f]
    -d <db_unique_name>      Unique name for the database
    -o <stop_options>        Options to shutdown command (e.g. normal, transactional, immediate, or abort)
    -f                       Force stop
    -h                       Print usage
$ srvctl stop instance -h

Stops the database instance.

Usage: srvctl stop instance -d <db_unique_name> {-n <node_name> | -i <inst_name_list>}  [-o <stop_options>] [-f]
    -d <db_unique_name>      Unique name for the database
    -n <node_name>           Node name
    -i "<inst,...>"          Comma separated instance names
    -o <stop_options>        Options to shutdown command (e.g. normal, transactional, immediate, or abort)
    -f                       Force stop
    -h                       Print usage
$ srvctl stop asm -h

Stops ASM instance.

Usage: srvctl stop asm [-n <node_name>] [-o <stop_options>] [-f]
    -n <node_name>           Node name
    -o <stop_options>        Options to shutdown command (e.g. normal, transactional, immediate, or abort)
    -f                       Force stop
    -h                       Print usage
$

Check if autostart for Oracle Restart is enabled:

$ cat /etc/oracle/scls_scr/$HOSTNAME/oracle/ohasdstr
enable

If it is not enabled, then enable it:

crsctl enable has

Check if ASM autostart is enabled:

crsctl stat res ora.asm

If ASM is not registered in GI, add it with:

srvctl add asm ...

Check if used diskgroups are registered:

srvctl status diskgroup -g DATA
crsctl stat res ora.DATA.dg

If they are not registered, add them with:

srvctl add diskgroup ...

Check if the database is registered and autostart is enabled:

srvctl config database -d ORCL

If it is not, register and enable:

srvctl add database ...
srvctl enable database ...

Finally make sure you define the used ASM diskgroups as dependencies:

srvctl modify database -d ORCL -diskgroup "DATA,FRA"

You can also check the listener:

srvctl config listener

If it does not exist, you can add with:

srvctl add listener ...

You do not need to start or stop anything manually with sqlplus, lsnrctl. Oracle Restart takes care of that based on the defined start/stop options and dependencies.