How do regulatory, allocative, distributive, and stabilization roles relate to government trade policy?

First, I think it is helpful to define these roles:

The regulatory role of government is its role in setting the rules--which actions are permitted and which are not.

The allocative role of government is its role in providing resources to support public goods, such as infrastructure and national defense.

The distributive role of government is its role in deciding to whom resources are allocated--in practice, this generally means deciding where to set the balance between free market outcomes and redistribution through taxes and transfers.

Finally, the stabilization role of government is its role in ensuring overall economic stability, particularly in terms of setting fiscal and monetary policy to reduce the harm and duration of recessions.

Now, to apply these to trade policy:

The regulatory role of government in trade is extremely important, and generally takes the form of trade agreements, which may set or remove tariffs, or require compliance with certain regulations such as pollution, human rights, or intellectual property.

The allocative role of government in trade is generally fairly minimal; most trade is conducted by private parties using private means (particularly sea vessels). One public good does become especially important, though: national defense. One of the main functions of a strong navy involves safeguarding trade against piracy and expropriation.

The distributive role of government in trade is somewhat indirect; governments do not generally decide specifically who will pay whom in trade agreements. A wise government will take into account the effects of globalized trade on local people and industries, however, and should use its redistributive powers to ensure the benefits from trade are spread as widely as possible, rather than concentrated in the hands of a few capital owners.

Finally, the stabilization role of government in trade involves currency exchange, including the decision of what sort of currency exchange regime (such as fixed, floating, or some hybrid) to maintain. Except in the case of a purely free-floating currency (in which case markets control that function), the government's stabilization role also includes responding to global market shocks in terms of devaluing or revaluing currency. This is surprisingly important; the value of a nation's currency in global markets can have profound effects on its terms of trade and overall economic prosperity.