<p>How (i.e. using which API) is the virtual keyboard opened on Symbian S60 5th edition? The documentation seems to lack information about this.</p>

You are right, this should obviously be a published API and it should be highlighted in the documentation. No such luck.
If you are using one of the platform native controls, the virtual keyboard will automatically popup when the user accesses a text-editing control.
If you are making a custom control, you need to deal with its selection by adding your own version of the virtual keyboard: make a new text-editing, window-owning virtual keyboard look-alike custom control with the right buttons. Reuse it accross all your applications. One day, Nokia will realize they have made an obvious mistake and make the API publicly available.
If you are using direct screen access, well, you wouldn't exactly expect the very s60-looking virtual keyboard to popup out of nowwhere. Again, draw a nice image on the screen to let the user know where the virtual keys are and react to pointer events. This is going to be less reusable unless you build a good amount of customization (background, button edges...) into it.
EDIT: Nokia may be relying on Qt to fix this issue. I would expect the control to be part of the current 4.7 version of Qt.