The reason may be inactive SMS module (ADMINISTRATION / SMS configuration. Activate module, set daily cash limit and your customers will be able to enter their phone number into the notification form.
Yes, you can control the number of SMS sent daily. Just use the SMS configuration available in ADMINISTRATION module. Set your daily cash limit there. System will not exceed the limit, so the messages not sent the same day will be sent on another.
Yes, such notification will be sent once.
Yes, just go to MODERATION / Content management for e-mails sent to customers / Letters sent automatically to customers / One-time notification of change in the availability of observed products and edit your message templates in any language you have active in your administration panel. You can also restore the default one at any moment. The same functionality is available for SMS - MODERATION / Content Management for SMS sent to customers.