No, there is no specific “peacekeeping and disaster relief” branch outside of the Coast Guard.
The National Guard is trained to do restoration of public order and emergency response jobs, plus their normal warfighting military functions.
As others have mentioned, there are units that are specially useful for civil restoration missions in the US military, such as MPs/SPs, sanitation, Public Affairs, construction engineers, etc. These types are present in both Regular and Guard commands, and the Army tends to assign units that are of special utility for disaster relief to the Guard – for instance, the PRNG has two units that do nothing but set up and run water purification plants, one of which has been in New Orleans, and had recently been in Iraq.
But even any sort of regular combat-arms or combat-support unit is useful in such a situation, for the reasons ex-Tank mentions.
To take the OP example, the 82d division brings to the disaster zone:
(a) a goodly of trucks and helicopters
(b) tens of thousands of highly motivated, physically fit people prepared to do stressful, strenuous work in unfavorable conditions
© …who are under an efficient centralized command structure
(d)…many of who are well armed and you never know if you’ll need it
(e) training in taking control of a sector of urban area and dealing with refugees
(f) expertise at the logistics of moving around large quantities of people and supplies in a secure manner
(g) internal Medical, Sanitation, Engineering, Public Affairs, etc. resources so it won’t tax the limited local ones and can actually help with some such work.
and this is not an exhaustive list.
Generally speaking, the philosophy when using military units in this role is to restore order and provide acute relief, but then as soon as viable, hand over the job to civilians, be they local civilians or contractors, and go back to the barracks to be ready to go fight.
Also, Wesley, the Army Corps of Engineers (which has a sort of “dual life” as a Corps within the Army and a civil-works Agency within the *Department * of the Army) has a big role here because because maintaining the Mississippi open to navigation (and preferably not flooding major urban, industrial or agricultural areas) is considered a national security interest and it comes within their strategic mission.