As far as I understand, these are the document coordinates of the center of the screen (ore the viewport, not 100% sure).

I want to use them and for now I am trying to place a dot at that point.

This however is failing very badly - probably due to unit conversion.
 

How is it that - with an A4 document (the default) zoomed to page width, my namedview is:

<sodipodi:namedview xmlns:sodipodi="http://sodipodi.sourceforge.net/DTD/sodipodi-0.dtd" xmlns:inkscape="http://www.inkscape.org/namespaces/inkscape" xmlns="http://www.w3.org/2000/svg" xmlns:svg="http://www.w3.org/2000/svg" id="namedview1" pagecolor="#ffffff" bordercolor="#666666" borderopacity="1.0" inkscape:showpageshadow="2" inkscape:pageopacity="0.0" inkscape:pagecheckerboard="0" inkscape:deskcolor="#d1d1d1" inkscape:document-units="mm" inkscape:zoom="1.4501687" inkscape:cx="396.85039" inkscape:cy="561.31402" inkscape:window-width="1862" inkscape:window-height="1016" inkscape:window-x="58" inkscape:window-y="27" inkscape:window-maximized="1" inkscape:current-layer="layer1"/>

and therefore cx and cy are respectively 396.85039 and 561.31402?

Try this instead

def draw_dot(self, center):
    from inkex import Circle

    circle = Circle()
    circle.set('r', 5)
    circle.set('cx', center[0])
    circle.set('cy', center[1])
    self.svg.get_current_layer().append(circle)

namedview = self.svg.namedview
draw_dot(self, namedview.center)

or maybe:

#Unit conversions
conversions = {
    'in': 96.0,
    'pt': 1.3333333333333333,
    'px': 1.0,
    'mm': 3.779527559055118,
    'cm': 37.79527559055118,
    'm': 3779.527559055118,
    'km': 3779527.559055118,
    'Q': 0.94488188976378,
    'pc': 16.0,
    'yd': 3456.0,
    'ft': 1152.0,
    '': 1.0,  # Default px
}


namedview = self.svg.namedview
draw_dot(self, namedview.center)

namedview_element = self.svg.xpath('//sodipodi:namedview')[0]
inkscape_cx = namedview_element.get('inkscape:cx')
inkscape_cy = namedview_element.get('inkscape:cy')

conversion_factor = conversions[self.svg.unit]

cx = float(inkscape_cx) / conversion_factor
cy = float(inkscape_cy) / conversion_factor

draw_dot(self, [cx, cy])