I don't know whether BitPay is profiting.
could they ran a trading desk and make profits off that?
I seem to recall Tony Gallippi denying that in an interview, I am not sure.
In that wallet snooping site we can see them collecting hundreds of small BTC deposits (many less than 1 BTC) and occasional 500-1000 BTC deposits (some identified as coming from KnC and other miners); and sending lumps of 500-1000 BTC, every day or so, to Bitstamp and other wallets. I suppose that they are not dumb enough to leave large sums on exchanges, so they must be selling at the same pace, at whatever price.
It is a pity that most bitcoin companies are privately financed, so they don't have to publish quarterly audits or other official financial statements. Bitstamp published a one-page report covering the year up to Oct/2013, containing the absolute minimum info required by UK laws (Shouldn't they be publishing the 2014 one now?). Last year, an australian bitcoin exchhange and mining operation published somewhat more complete financial reports. The current state of MtGOX spoils was published by the bankruptcy trustee. Have there been any others?