I've never heard the 1812 Overture associated in any way with the Fourth. 

Of course I've only lived in the US for sixty-odd years, so maybe I missed it.

I wouldn't say Russia won the battle of Borodino, even if the Russians think so. General Kutuzov didn't _lose_ it, though, so the Russians came off a lot better than they might have expected. The French were left in possession of the field, free to continue their advance on Moscow, so in conventional terms that would usually be considered a French victory. The bodies were left where they fell; the French actually found a survivor when they crossed the battlefield a year later, during their retreat in 1813. Had the Russians won and ended up in possession of the field they would have buried the bodies. Russians in those days were very religious people and felt obligated to treat fellow Christians to proper burials.