Every country (or other jurisdiction) can make its own laws and regulations about how Bitcoin can be exchanged for dollars (or anything else) in their jurisdiction, just like they can make their own laws about taxes, traffic, etc. So your question as stated maybe has hundreds of answers.
Since you mention US dollars: in the United States, my understanding is that most regulations on businesses that exchange Bitcoins come from the fact that they are considered money services businesses (MSBs). The US Department of Treasury, through its FinCEN agency, makes and enforces regulations on MSBs, as authorized by laws created to fight money laundering. These regulations require MSBs to do various things, such as gather personal information on their customers, report suspicious transactions to the government, and so on.
I don't know much about gold markets, so I can't really address that part of your question. I would think that in the US, large-scale gold market participants would fall under the purview of the US Commodity Futures Trading Commission and be subject to regulations roughly analogous to those which the Securities and Exchange Commission enforces on stock traders and brokers. My impression is that these regulations are substantially more complex than those for MSBs. I don't think this would apply to small-scale operators like jewelry stores, but they may have other regulations to obey, perhaps imposed at the state or local level.