Am I able to get the current call history with the call number and DateTime details from the phone with the Mango API? I think it is not possible with the current one.
There is no API to retrieve Call History. This is mainly due to privacy concerns and protecting the user from nefarious applications behaving badly on the device.
You will notice that among the API is a collection of Launchers and Choosers. One of which is the PhoneNumberChooser Task. This particular task launches the Contacts application and allows the user to physically select a contact number to dial from your application. You'll notice this requires a manual interaction from the user. Many of the data retrieval APIs require manual selection from the user, once again to protect the user from applications doing bad things. For example, trolling your contact list in the background and sending that data to a web service without the user's knowledge.
Special care was given to the API's to protect the user from these types of scenarios.
If you have feedback on the need for an API to provide a Call History Chooser, submit your feedback to http://wpdev.uservoice.com.