I wondered if there was a "good" (I mean that in a non-technical way) consciousness?
Consciousness exists as a taker of objects. Consciousness works together with perception to cognize an object in the cognitive process.
Consciousness is not a thing. The problem is that we are forced to talk about it in conventional terms using nouns. That can give the impression that there is "something" there but there is not.
Since there is nothing there, there is nothing to superimpose qualities onto such as "good, bad etc.". The act of superimposing qualities such as "good", "bad" etc. onto objects are taught by the Dalai Lama in his book "How to see Yourself As You Really Are" in the chapter "Identifying Ignorance" on p. 45-49 and the chapter "Why Understanding the Truth Is Needed" on p. 63 where he writes:
"The supreme scholar-practicioners i India - Nagarjuna, Aryadeva, Chandrakirti, and Dharmakirti - understood that the truth cannot be realized without seeing that we superimpose on people and things a status of solidity and permanence that actually is not there. The emptiness of that false superimposition must be understood, and to do this they analyzed phenomena through scripture and through reasoning".
"When ignorance is overcome, you will have uprooted the mistaken beliefs that superimpose on objects qualities such as beauty and ugliness beyond what they actually have".
I have made a highlights in this above sections.
What sort of cognition is the cognition of extinction?
I am not sure i understand your question correctly. I will try to answer as best as i can.
When dealing with conditioned phenomena both physical and mental they all follow a specific process. That process is "arising, presence, dissolution".
When doing insight meditation one can observe this process clearly. The dissolution/extinction part of phenomena happens e.g. when the abdomen has reached full extension on the inbreath and then begins to fall. That is the start of the dissolution step. When the falling of the abdomen has finished then the dissolution step has finished and one can clearly see the impermanence of the breath. This dissolution step can be seen in all phenomena and when observed it will grant access to understanding the impermanent nature of conditioned phenomena.
This is taught by Ven. Mahasi Sayadaw in his book
"A Discourse on the Anattalakkhana Sutta" in the chapters "Refuting A Self Apart From the Five Aggregates" and "Why It Is Called Impermanent" both on p. 85-90.