Transparency Issue With Placed Graphic CS6

When I place a graphic into Illustrator (in this case from Photoshop, a black and white image of a glass), I trace it without issue, and then want to place it as the top layer on a black square.

First issue doing this, is that the placed item is in a frame that is not transparent, therefore there is a white box surrounding the glass, which stands out due to the black background.

Ok, to address this, used the advanced setting for trace to ignore white (windows>image trace: tick 'ignore white'). The graphic now has a transparent frame, but the white parts within the image itself are now also transparent.

What is the best way to resolve this? I was thinking of just making a copy of the now transparent object, fill this copy of the object so it is solid white, and put it as a layer beneath the fully transparent one.

This does not sound like the correct way to approach this, and besides, have been unable to create a solid white copy of the original in any case.

Any assistance greatly appreciated with the best practice for what I am trying to achieve, and also would like to know how to create solid white filled copy of the graphic. The graphic is enclosed within black outline with white on the inside and a black outline, so nothing overly complicated.

There are always problems using the Trace functions! Can I suggest that you use the "Shades of Grey" profile when using Trace. Then use Expand then ungroup your trace results from the object menu to convert it into many vector shapes. Then use the Layer Panel to find the shape background that is white and remove that. You can go on and remove other artifacts that are unwanted as this is the problem with the tracing results.
If I was in your shoes and if the glass was a relative simple shape I would simply draw it an manually give your shape the Opacity that you require and this shouldn't take long. Please remember that you have Gradient fills that you can use and since you have CS6 then you also have Gradient Strokes which are very useful in this area if you use one of the Artistic Chalk and Charcoal brushes from their library and I would also give these strokes opacity. Also you have the image that you want to copy so you can sample colours from that when choosing your gradient.

Expand the trace and delete the parts of it you do not want.

Please include a screen shot of what you have on the page, and likely with the layers panel open and expanded to see the objects as well.

Mike