Is it safe to copy DB files directly from an LVM snapshot?
Yes, it is safe. There is a tool http://www.lenzg.net/mylvmbackup/ that uses LVM snapshots to take backups of MySQL.

From InnoDB's standpoint a backup copy taken with LVM snapshots looks like as InnoDB has suddenly crashed. When you start MySQL from the backup copy it will start crash recovery. To save some time it's better to do the crash recovery immediately after the backup is taken.