Some answers:
1. You can connect the Visual Studio to a database using for example Server Explorer:
http://msdn.microsoft.com/en-us/library/cc440724(v=vs.90).aspx[
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2. No, LINQ is a mechanism you can use to for example query data from SQL Server (or from objects or XML etc). The data is often stored in a database such as SQL Server. So LINQ isn't an alternative for a database.
3. It's not actually part of Visual Studio but part of .Net framework. Perhaps this would explain more:
http://msdn.microsoft.com/en-us/library/bb308959.aspx[
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4. The link in bullet 1 should answer also this.