How to draw an $405^\circ$ angle? In a math test a question was to draw a $405^\circ$ angle. Is it formally correct to say draw an angle as I think that in geometry, an angle has just some formal definition. So what is the connection between the formal definition and the drawing? And how do one draws angles over $360^\circ$?

It depends on how you think about angles.  You can either agree that $405^\circ$ is exactly the same as $45^\circ$.  -- This is how mathematicians usually think about it.  Or you can think about it as $1$ complete rotation ($360^\circ$) and then an additional $45^\circ$.  -- This is how engineers usually think about it.
The way you draw it depends on which of the two ways above you think about it, but the angle should start on the positive $x$-axis and end on the ray positioned at a $45^\circ$ angle counterclockwise from that position either way you think about it.
