Is it permissible for employees to skip their lunch or break periods in order to leave early? Can employees if paid for it, have an “on duty” meal period?

No. The California Labor Code § 226.7 invokes penalties against the employer if it fails to provide a meal or rest period. There are two exceptions to this rule: (1) if work hours do not exceed six hours in a day, then the employee may waive, in writing and with the employer’s consent, the meal period; and (2) for shifts exceeding ten hours in a day, the second meal period may be waived in writing by the employee, again with the employer’s consent. The exception addressing a second meal period does not apply to Wage Orders 4 or 5. Only employees in the healthcare industry may waive the second meal period for shifts exceeding twelve hours.

The answer to the second question is “no” also. Employees can have an on-duty meal period in limited circumstances – only when (a) the nature of the work prevents the employee from being relieved of all duty; (b) the employee and employer agree in writing to an on-duty meal; and (c) the employee is paid for the meal period.