How to access Google Maps API v3 marker's DIV and its pixel position?

I don't really get why would you want to get specific div for marker? If you want to display tooltip then all you need is pixel position of markers anchor (and knowledge about size of marker and placement of anchor), not div element. You can always trigger opening and closing tooltip by hand when event occurs on google.maps side.

For getting pixel position of anchor of given marker you can use this code:

var scale = Math.pow(2, map.getZoom());
var nw = new google.maps.LatLng(
    map.getBounds().getNorthEast().lat(),
    map.getBounds().getSouthWest().lng()
);
var worldCoordinateNW = map.getProjection().fromLatLngToPoint(nw);
var worldCoordinate = map.getProjection().fromLatLngToPoint(marker.getPosition());
var pixelOffset = new google.maps.Point(
    Math.floor((worldCoordinate.x - worldCoordinateNW.x) * scale),
    Math.floor((worldCoordinate.y - worldCoordinateNW.y) * scale)
);

In pixelDistance you get offset of specific marker anchor counted from left upper corner of the map (and you can get it's position from map.getDiv() div). Why it works like this (or is there a better way?) you can read in documentation of google maps overlays.

var overlay = new google.maps.OverlayView();
overlay.draw = function() {};
overlay.setMap(map);

var proj = overlay.getProjection();
var pos = marker.getPosition();
var p = proj.fromLatLngToContainerPixel(pos);

You can now access your pixel coordinates through p.x and p.y.

FOLLOWING ADDED POST COMMENT:

The downfall of the overlay projection is that until it your map canvas finishes loading it isn't initialized. I have the following listener that will force whatever method I need to trigger when the map does finish loading.

google.maps.event.addListener(map, 'idle', functionName());

In the mean time I use the following check to avoid any errors before it does draw.

if(overlay.getProjection()) {
  // code here
}

One thing to remember when using MBO's code: When the map tiles are repeated, map.getBounds().getSouthWest() returns "-180" independent of the map's position. A fallback I'm using in this case is calculating the pixel distance to the center instead of the upper left corner, since map.getCenter() seems to return the currently centered point in any case. E.g. (using jQuery):

// Calculate marker position in pixels form upper left corner
var pixelCoordsCenter = map.getProjection().fromLatLngToPoint(map.getCenter());
pixelOffset = new google.maps.Point(
    Math.floor((pixelCoordsMarker.x - pixelCoordsCenter.x) * scale + $(container).width()/2),
    Math.floor((pixelCoordsMarker.y - pixelCoordsCenter.y) * scale + $(container).height()/2)
);