I am running a site right now with a quite big MySQL database. Now, I want to create an app. I will need to use obviously a database de to the fact my data are already there.
Thus,
1) Should I keep using the MySQL server and my iOS app will connect to this MySQL serve for getting data?
2) is there any problem if I use the MySQL server ? Security issues maybe?
3) if I have to change the MySQL server, what database infrastructure I need to build and work with?
I am totally newbie on iOS apps. And now I planning to face any issues my iOS app will have.
Since you are running a site with MySQL I think a lot of your questions have been answered. You already have seen how your MySQL database performs in a production environment. Unless you believe that your iOS user base is going to be much larger or perform very different functions you currently have valid performance benchmarks.
In terms of how it relates to the iOS app, you can build an API, or make calls to https pages on your website that will return the information required for your app. For information on how to do this, check out this AFNetworking tutorial.
In the end, there might be some reasons that your current database isn't the right choice for your app. But since the app won't be interacting directly with the database, you can change it out later and you will only have to integrate the database with the interface and not change the iOS app at all.