Let's take it one by one.
Microsoft SQL Server Management Studio is a gui tool to manage most of the functionality included with SQL Server any edition. Have a look here: http://technet.microsoft.com/en-us/library/ms174173.aspx
]. Well, have a really deep look, by reading most if it, if possible.
This should be included in the Developer edition, and you should be able to select it when installing. But I encountered people saying that they could not manage to install the tool directly. But no problem, you can install Management Studio Express separately. I suggest you install the 2012 version which can be downloaded from here: http://www.microsoft.com/en-us/download/details.aspx?id=29062
] (click on download and check only the Management Studio x86 or x64 in the list)
And finally: no, SQL server is not included in Visual Studio.