Good morning Jeffery,
1) Is is the employee's responsibility to as for vacation time and/or sick time to be used to cover this time off or should it be automatically be used? You may automatically deduct for sick leave if they have it accrued when they are out with an illness or injury. As for vacation time, if your company policy holds that vacation time may be forced when an employee is out sick and has no further sick time left, then you may do that as well. The same right is afforded larger employers who have employees go out on leave under the FMLA or the CA FRA.
2) How long must I give the employee to comply to the request for physician's release? You can give them as little time, or as much as you choose. If they have used their sick leave up, and because you are not required to force them to take vacation time, you could demand that the doctor's note be submitted no later than their last sick day, and after that they are subject to termination.
3) Can the refusal up to this point to comply with request for physician's release (since injury was reported last Friday am) be substantial time to warrant employee's termination? As the employee handbook clearly gives you the right to ask for a note---and the employee has refused unless you give in to their specific demands---that constitutes insubordination for which you may fire them and they will not be eligible for unemployment.
4) What time frame, if any, does the labor law state for employee to comply to request? The CA Labor Code does not set a specific response time for providing this information to employers. Again, you are a small employer---less than 5 employees---and the typical regulations don't apply to you. While the FMLA has time requirements, you are not obligated to follow them.
I think the overriding issue you present here is the complete insubordination of the employee---and that you never have to tolerate.
You may reply back to me again, using the Reply to Expert link, if you have additional questions.
I wish you the best in your future,