As a result, my website was down for few days.
After further investigation, it turns out that they didn't export and import mySQL database correctly. Some of the tables didn't get the auto_increment attribute on the primary key so it was failing when a new record was trying to be added.
My lesson learned from this? NEVER trust your hosting company when they are doing any sort of upgrade. You should back up your database and the entire website content before they do anything.
In fact, I'm not comfortable with their so-called daily back-up. I would back up the entire site, including the database, once in a while just in case.
I'm also planning to move my hosting plan to a dedicated or virtual server so that I would have the total control of the server.
My website seems to be up and running now after fixing all database issues. Please let me know if you find any problems. Thanks!