Can we install configuration profile automatically with MDM, instead of using safari and settings?
Yes. You can. It will be invisible to a user (except, if a configuration profile requires some user interaction).
Can we update apps in customers' devices automatically without using app store?
For non supervised devices, you can install/upgrade an application, but a user will be required to accept it.
For supervised devices you can install/upgrade apps silently (starting from iOS 7).
There are some features in MDM, does it mean we can access data and control customers devices without notice?
Generally no. MDM doesn't give you an access to customers data, except a case if application is specifically designed to return data through MDM.