does someone have integrity if they choose to be fair and honest only in certain circumstances? Does that make you them trustworthy

Yes the meaning of integrity is the following

the quality of being honest and having strong moral principles; moral uprightness.

Someone who has integrity is someone who acts morally and according to fair principles, even when these might go against his personal interests. A person who has this quality will generally conduct himself in this manner at all times, not only in particular cases. Therefore, if a person chooses to be fair and honest only in some circumstances, then he should not be considered to have integrity. Moreover, as his behaviour is irregular and advantageous, this means that he should not be considered to be trustworthy.