Hi Sean, the best way to get rid of that edge (which again is not really necessary, it should generally be ok to have it such a small distance away it is well within tolerance), would be to do some lower level operations than booleans. I'm not exactly sure why the extrude of that one face doesn't produce any result, but that probably does indeed have something to do with this edge you're worried about - when you extrude a face it is really a batch operation of internally generating a non-attached extrude followed by a boolean union between that extrusion and the main piece. Like I mentioned earlier, booleans between pieces with overlapping surface areas do tend to work better when they are very precise in the overlapping region.

So anyway to remove your pernicious edge ;) try the following process - select the 2 adjoining faces to it, the top and side wall faces and delete them. Now that edge is gone and you can construct a better one to take it's place. You'll do this by using Draw curve > Line, and placing a new line between the 2 corner points, make sure Object Snap is enabled and you see an "End" snap tag show up for each point of the line you place, that way you know the new line is snapped on to ends of those other existing edges.

After your new line is placed, select the new line and the 3 open edges of the top square, and the circle edge in the middle of it. So now you have 5 objects selected, one line curve you just drew, 3 line edges on the main solid, and one circle edge on the main solid. Now run the Construct > Planar command to build a trimmed planar surface through those selected objects. You can now clear the selection and select the line you previously drew and delete it, you don't need it anymore.

Now select the new planar surface you just constructed, and also the main object. Now use Edit > Join to glue those together. Now you need to rebuild the side wall as well, since that's a planar opening bounded on all sides by edges there is a shortcut for doing that, just select the entire object and run Construct > Planar and it will build a surface there and automatically join it. Construct > Planar can be used in 2 different ways, when used with a curve selection it will build an individual planar surface through those curves. When used with a joined surface object that has open edges, it will look for any sequence of unjoined (so-called "naked" edges) in the object and build planar surfaces through those and automatically join it in.

It looks like there may be another couple ones like that around because I'm still seeing that a face extrusion of that vertical side is not behaving well. I tested it by selecting the edges around that face and duplicating them as curves and then going to the Front view and trying to squish it down and seeing "flat" snap engage (see here for a demo of flat snap: http://moi3d.com/forum/index.php?webtag=MOI&msg=3378.4) - the thing is flat snap will only engage if the objects being squished were not already very tightly planar. But if you do want to extrude in that direction it seems to be enough to duplicate the edges, flatten them with flat snap, then extrude those and boolean union that result to the main object.

Hope this helps!

- Michael