The database registers its service name(s) with the listener when it starts up. If it is unable to do so then it tries again periodically - so if the listener starts after the database then there can be a delay before the service is recognised.
If the database isn't running, though, nothing will have registered the service, so you shouldn't expect the listener to know about it - lsnrctl status or lsnrctl services won't report a service that isn't registered yet.
You can start the database up without the listener; from the Oracle account and with your ORACLE_HOME, ORACLE_SID and PATH set you can do:
sqlplus /nolog
Then from the SQL*Plus prompt:
connect / as sysdba
startup
Or through the Grid infrastructure, from the grid account, use the srvctl start database command:
srvctl start database -d db_unique_name [-o start_options] [-n node_name]
You might want to look at whether the database is set to auto-start in your oratab file, and depending on what you're using whether it should have started automatically. If you're expecting it to be running and it isn't, or you try to start it and it won't come up, then that's a whole different scenario - you'd need to look at the error messages, alert log, possibly trace files etc. to see exactly why it won't start, and if you can't figure it out, maybe ask on Database Adminsitrators rather than on Stack Overflow.
If the database can't see +DATA then ASM may not be running; you can see how to start that here; or using srvctl start asm. As the documentation says, make sure you do that from the grid home, not the database home.