If all employees are part-time (less than 20 hours week), am I required to carry worker's comp. and unemployment insurance? What if I don't?

Yes, you have to carry both. worker's comp compensates employees for on-the-job or job-related injuries; while an employee's hours, and therefore his/her earnings, will affect how much he or she can get, it does not affect the requirement to have it. (You have to have worker's comp for all employees other than employees who are also owners of a business, such as partners in a LLC; they may waive, with the proper paperwork, having worker's comp.)

Similarly, you have to pay unemployment based on a percentage of worker earnings--it is essentially a "tax" the state puts on your business, and you have to pay it usually even for workers who do not themselves work enough to qualify for unemployment.

Severe penalties and interest (on back or unpaid amounts) may be assessed on employees who do no meet these requirements.

Part time status is usually grounds for not providing workers all the benefits (e.g. health, vacation, etc.) that full time workers get, but does not affect obligations imposed by the government. For more information and to confirm/clarify exactly what you have to pay, contact your state labor department.