Regardless of stink or soil, the rules are that you cannot open the animal's carrier while it's on the plane, not even just a couple of inches to pet it. I do see people breaking this rule all the time but if the FA's see, they reinforce the rule. It's sad and a bit cruel to think of any animal being confined and sitting in its own mess for any amount of time but that's one of the risks you take when flying with an animal. Before I get flamed for being anti-pet, I'm not; I choose to leave mine home most times, but I couldn't avoid flying with cats twice this year, so I know it's difficult.
But the carrier must remain closed, no exceptions, because animals can be very unpredictable in a stressful situation; it is for their protection as well as for the protection of everyone on the plane, as many jurisdictions have a one-bite-rule. A dog or cat that bites just once is taken by animal control and put down. And I would rather have my dog crying and soiled than take the most remote chance that she would bite someone out of fear, then be put to sleep.
That said, the passenger should have asked for a blanket and some air freshener, and left the dog in the carrier till landing. She should not have even attempted a cleanup in flight.
I don't usually agree with sedating animals for flight as there can be some bad reactions to the sedatives, especially among cats who can occasionally end up more panicked and wound up than without a sedative. If you feel you must sedate your animal before flight, do a few "test runs" at home, in a controlled environment, to see how he/she responds to the sedatives (much like taking Ambien for the first time on the ground instead of on your transpac flight). It's absolutely essential that the animal be "empty" before flight.