I have experienced both sides of the 6 inch debate. In my three beds that are open to the tilled bare dirt beneath them, yes, 6 inches is most certainly enough. In the rest of my garden for which the boxes are lined with landscape fabric or cardboard, 12 inches seem to work best. I had 6 inch boxes with cardboard last year and they flopped. I just put additions on them that you can see with the link below.

Here is a link that might be useful: Raised - raised beds

I would go at least a foot deep if you are going to put a bottom on them - or grow them on some kind of material that the plants cannot grow into (like concrete - since I have some boxes on my driveway).

If you are going to put the boxes in your lawn, I don't see any point of putting landscape fabric between the ground and the bottom of the box - this just hinders the plants from having a deep root system and allowing for additional nutrient up-take. Even so, nitrogen and other nutrients will be able to make its way under the fabric - but roots can't.

Here is a link that might be useful: BsnTech Gardening Blog