Jimmy's trial is not a trial. It is a hearing by the Bar Association.
In an earlier episode, Chuck specifically mentions that a hearing does not have the same level of standard (for both evidence inclusion and witness testimony) that a normal court case would have.
From the script:
Kim - There's no way you were gonna let him destroy the only copy of that tape, so you made a duplicate - it's the first thing you did.
Howard - Kim, this is not how we do discovery.
Chuck - Actually, Jimmy destroyed the duplicate. [Howard wants to stop Chuck from talking] Ah, it's fine, Howard. [Breathes deeply] The original is under lock and key, and in due time, it will be put into evidence for your review. So, of course, file whatever motions you'd like.
Kim - I will. Count on it.
Chuck - But, Kim, you should be aware because I believe this will be your first disciplinary hearing uh, the Bar Association's standard of proof is far more lenient than what you're used to [in a normal court case]. Motions aside, that tape will be played.
Normal trials have a jury made up of "ordinary" people. Many strict evidentiary rules are upheld because "ordinary" people are no law experts.
However, the hearing is held exclusively by those who practice law, so there is less need to make the legal precedings understandable to those who do not practice law.[footnote]
Furthermore, this is not a court case. Jimmy is not looking at any jail time or fine. This hearing is much closer to a work-related meeting where they discuss suspending an employee or not (although the Bar Association works on a level higher than a single law firm).
It's also relevant to consider that the hearing is not trying to ascertain Jimmy's guilt, but rather whether or not his actions are in violation of the code of conduct (which is upheld by the Bar Association).
The "grey area" admissions you hear (that you would not hear in a normal court) are allowed here, because to judge Jimmy's actions, the panel of judges needs to know Jimmy's emotional state and what his intentions were.
Edit - [footnote]
This is a bit too roughtly put; the jury is not the only reason why certain legal precedings exist. However, for the purpose of your question, it can be considered correct enough, in my opinion.
My comment below:
[..] Jury bias is a major reason for excluding evidence/testimony before it is presented, compared to not using it after it is considered invalid. A judge can see the relevant from the irrelevant, but a jury is not expected to be able to.
There are many instances (at the very least in TV and movies) where a court case's "obstructional rules" (for lack of a better term) are ignored when they serve no functional purpose (e.g. when all parties involved are considered to be capable lawyers who don't make legal mistakes) and it would otherwise obstruct the plot's flow for the viewer.