It is possible, but not by using the 'Simple Fill Symbol'.
Method 1
Create a transparent image for the colour of your choosing. You can do this in GIMP by creating a new image and setting the Opacity (on the Layers tool palette) to whatever you like, and saving it as a PNG file. Other image editing software will work fine as well. The blue image below is an example. The image can be any size, 10x10 pixels will work.
In ArcGIS, use the 'Picture Fill Symbol' method to fill your polygon, and choose the PNG file as your picture. Choose your outline as normal and it won't appear transparent! The only drawback is that you have to re-edit your PNG to change the transparency/opacity of the fill.
Additionally, you can use this same method with 'Picture Line Symbol', if you'd like your boundary to be transparent too. You could have a play around with the images in Photoshop (or something similar) to adjust the transparency, as you see fit.
Method 2
An alternative method, granted it won't give you the exact appearance which you were after, would be to use the 'Marker Fill Symbol' method. By using a super tiny symbol and having a play around with the symbol separation, you can create a fairly transparent fill of a sort. See example below.
Both of these methods will stop you from having to use multiple layers.
You can use Advanced Editing toolbar with Construct Polygons tool for that (available to ArcGIS for Desktop Standard and ArcGIS for Desktop Advanced only).
On the picture to the right, the selected lines that form a triangle are used for constructing a polygon yet an existing large polygon is taken into account and just two small triangles are created. It is important to make sure the lines are closed, otherwise you cannot generate polygons. In case they don't and you have many features to go through, you might consider using Integrate GP tool to "snap" closely located vertices of lines to form closed geometries.
Best Answer
There is an example in the SDK samples http://resources.esri.com/help/9.3/arcgisdesktop/com/COM/ExtendAO/VertexLineSymbol.htm
Description This project provides a custom symbol to draw a line and its vertices. Simple custom functionality is provided to alter both the symbol used to draw the basic shape of the line, and the symbol used to draw its vertices. A property page is also provided to allow users to edit the properties of the symbol using the user interface.