I trust the EPA more than I do the Alaska Department of Natural Resources, and considering this is on state land, DNR takes a precedence on much of it. As for the EPA's and Army Corp of Engineers role, much of it is pending on a USSC decision that is due sometime soon. Which is based around the concept that a mine near Juneau wants to dump tailings into a lake. This will kill every organism in the lake. But the decision will be based on whether this is acceptable because apparently the lake will rejuvenate over a period of a few decades.
http://aprn.org/2009/01/12/supreme-court-hears-kensington-mine-toxic-waste-case/
But as for "doing it right" there simply isnt an economically viable way to build this mine without wiping out a minimum 250,000 salmon, that would otherwise be there every year. This is because the water table would have to be lowered to several thousand feet below the current level at the surface, that will effect a massive area surrounding the potential mine.
Then of course there is the tailings embankment which will basically cover the existing streams. It simply cannot be done without having a serious impact on the rivers, and the fish that live there, which are necessary for commercial and subsistence use.
So to simply answer your question, as for if I trust the various government agencies that would apply to this mine, I first and foremost dont trust Alaska DNR. Thier purpose is to develop the various resources of Alaska(minerals-oil and gas, and timber, etc.) but when it comes to mines they have had some dramatic recent failures. The Illinois Creek Mine, an open pit mine using cyanide leach methods, went backrupt shortly after opening and the state had to step in and operate the mine until enough funds were made to reclaim the mine. Then just recently there have been some major problems with the Rock Creek Mine near Nome. Which basically has shut down, and the state will probably also have to step in there because in each case the reclamation bonds required havent been anywhere near the actual cost of mine reclamation.
There are a few other examples of mines having problems and not coming anywhere near the EIS(environmental impact study) claims that are made. A perfect example would be the Red Dog Mine near Kotzebue, which has literally had thousands of water quality violations, but is still in operation, and has been subject to a fairly major lawsuit over this issue. Then a small cyanide leak at the Fort Knox Mine near Fairbanks.
So the track record is very questionable when it comes to mines that are to undergo the very same permitting process as Pebble would.