No.  You should be able to write your customers numbers in a way that makes sense to you.  Appointment Reminder will be smart enough to remove any brackets, dashed or hyphens.

If you want to send a SMS reminder you just need to enter the mobile number of your customer.  You should however always enter the area code.  Country code is not required.

*US Customers Only
For clients in the US in order for Appointment Reminder to distinguish between a SMS reminder and a phone call reminder you need to add ph: in front of any number where you want to send a phone call reminder.