SQL Server Agents is a Microsoft service that allows you to automate certain tasks (or at least that is the use I give) to use your SQL Server instance that service is not necessary if you set your instance (SQLEXPRESS) it's automatic and runs perfectly.
Regarding your problem, SQL Server Management Studio works because you entered the data correctly. According to the ScreenShots you published the error you have is that you have not entered the name of your SQL Sever instance correctly. Apparently your server name is EDUARD-PC \ SQLEXPRESS so that same you must enter in the Visual Studio. In Visual Studio I see that you say that your instance is "." and that's why he sends you that error. Try changing where it says Server Name and type EDUARD-PC \ SQLEXPRESS
I was doing tests on my computer about connecting Visual Studio 2015 with my Local Instance of SQL Server 2014 using the form that you use (that is "." for connection locally) and neither in SQL Server nor in Visual Studio has allowed me connect me.
SQL Server 2014
Visual Studio 2015
I would recommend that you put EDUARD-PC \ SQLEXPRESS in the VS 2015 to connect. You just try maybe and it works for you.