Yes a dog could kill a badger if the badger is alone and the dog is a large enough dog.
Most badgers will try to avoid the dog or will squirt the dog with a foul smelling secretion from the badgers anal glands to ward off the dog.
But sometimes the dog still gets into a fight with the badger and the dog when large enough will win the fight.
I've seen a large Pit Bull kill a badger before so it's possible for the large dog to kill the badger.
However if it's a smaller dog and the smaller dog gets into a fight with the badger then the badger will likely kill the dog instead or at least severely injure the dog.
The reason a badger would be out in the day is that the badger is likely foraging for food when the food is scarce.
Most badgers are mostly out at night looking for food or just out and about but it's normal to see a badger or badgers out in the day as well.
It doesn't mean anything is wrong with the badger if it's out at daylight hours.
I've seen a few badgers out at daylight when camping before and they were just out looking for food and collecting food.
Although a badger is a pretty small animal the badger is pretty strong and can usually defend itself easily against other large animals.
I've seen a honey badger come up against 2 lions and the honey badger actually won the fight.
The badger also can release a foul smelling secretion from it's anal glands which stinks and helps to ward off the attacker of the badger.
The reason badgers stink is because they use a foul smelling secretion that they release in self defense.
The badger and particularly the stink badger has foul-smelling secretions that they expel from anal glands in self-defense.
So when you smell a stink badger or a badger that stinks it's because of the foul smelling secretions from their anal glands.
The smell can linger for days but eventually the smell will go away if the stink badger does not need to release that secretions for awhile.
But as soon as the badger needs to release that secretion from it's anal glands the badger will stink again.