Do opposite spells automatically cancel each other or does the casted spell have to specifically target the active spell to act as a counterspell?
For example, a demon casts deeper darkness on itself, adjacent to a character with an active continual flame. Does the deeper darkness spell automatically cancel the continual flame, resulting in normal light conditions? Or does the demon have to specifically cast his spell as a counterspell to continual flame? What if the character's active spell was light instead of continual flame?
Another example, the party walks into a room with haste active. Recognizing this, the BBEG casts slow on the party. Is the party's haste automatically negated, removing the buff? Do the PCs get a save against the slow to see if their haste is removed? Or (which I don't think happens, but could be wrong) the party loses haste and is affected by slow if they fail their save?
I think these spells mention whether or not they can counter, can dispel, or do anything else in their description. I'm not aware of any external rule that says otherwise.
Spells like slow and haste can be used to counter or dispel each other because they say so in their description. Darkness can be used to counter and dispel light spells of an equal level or lower. Non-magical light does not increase the light level in the area of a darkness spell, nor does any light spell unless it is a spell level that is greater than the darkness spell.
By the way, darkness is an evocation [darkness] spell. Continual Flame is a evocation [light] spell. Apparently this relationship was important enough to include a mention when declaring what school they belong to.