I think the issue with the SMS is resolved. However, now there seems to be a problem with scheduling appointments at all. After the schedule is completed and submitted, a positive validation message is shown, yet the appointment does not appear in the appointments section and there is no SMS sent notifying of the scheduled appointment.

Most likely cause of a failure after validation is telling ABPro to use Google Calendar but not having the GData installed or having a problem with the login Google credentials.

Turn off 3rd party calendars, ABPro Configure screen | Calendars tab, set to 'None'.

I don't know what's happening now. I set the calender from Google to none. I know the Google credentials are correct as well as the calendar link. Still am not able to process a schedule. Everything is accepted and the scheduling is validated, but it is not showing up in the appointments tab. No SMS is being sent now, notifying me of the booking.

Turn OFF SMS, turn OFF Google.

There are many settings in ABPro.It is best to start simple then enable additional complexity incrementally, that way you can see when something is causing problems.

Your SMS may be causing the booking to abort. Look at the ABPro Error log to see if there is any info there.
ABPro Control Panel | Error Log.

Turned off the SMS and Google. Tried to book an appointment again and still does not go through. Rather than going through and giving the message that the appointment has been booked, it returns to the beginning of the booking process. Looked in there error log and there is nothing there.

I have never heard of the behavior before.
Something very strange on your site.
Do you have SEO enabled?
The only thing I can think of is that SEO (or .htaccess) is messing up the AJAX call URLs.