A:
Credibleness sounds very unnatural, and I don't think it exists.
Credibility is the quality that something is believable or trustworthy. A few examples:
When the news of the scandal broke, he lost all credibility.
(He was trusted and believed by everyone until they found out about his scandal)
When writing a research paper, it is important to use trustworthy and credible sources.
(The adjective form of credibility is credible, and it just means "has credibility.")
New evidence has called the credibility of his testimony into question.
(His testimony was believable and trustworthy before, but in the light of new evidence, it is now clouded by doubt.)
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Credence is the belief that something is true. It is not a quality like credibility. You "give" or "lend" credence. It sounds quite formal, and is rarely heard in colloquial situations.
Even though he swore he told me the truth, I could not give any credence to my student's story that his dog ate his homework.
(Basically, I did not believe his story at all.)
The accounts of multiple witnesses lent credence to the suspect's version of events.
(The accounts of multiple witnesses supported/corroborated the suspect's version of what happened that day.)
Give credence is like "believe to be true"
Lend credence is like "support or increase the likelihood that something is true"