Is there a way in Intellij IDEA to see the name of the method the current line belongs to?

You can use View | Context Info (Alt+Q, Ctrl+Shift+Q on Macs). It will display a pop-up on the top of the editor with the current context information (class/method signature).

IntelliJ 2018

This is shown by default at the bottom.

Unfortunately, the method is shown only by name (not including the parameters). If a method is overloaded you won't know for sure where you are.

If you want to move it from bottom to top, go to File > Settings... > Editor > General > Breadcrumbs > check Top:

In the structure panel select the "Autoscroll from source" option. This way when you place the cursor inside any method the structure panel will show which method you're in.