The Trial (possible spoilers)

My sense is that the trial scenes are highly truncated. Obviously, a murder trial would last a number days, and involve a great deal of evidence. The point of those trial scenes is really to show Louis’ more arrogant and complacent side. After all he knows he’s innocent, which is the whole irony of the end of the film.

I doubt a denial of his relationship with Sibella would work. As Sibella herself observes in the film, a married woman calling on a bachelor would have been noticed, particularly at Louis’ apartment block. Lionel may not have known about the affair, but he certainly knew Louis had feelings for Sibella, as did everyone, presumably. The affair wouldn’t be hard to prove.

Also, Lionel is one of the bank’s customers, he’s in trouble, Louis’ secretary would have booked an appointment (or at least known where he was). So it would be fairly easy to place Louis at the scene around the crucial time.

So you’ve got a credible motive, and you can place him at the scene of the crime. That coupled with Louis’ shocking performance on the stand might have been enough. Enough for suspension of disbelief, anyway.

Since the trial was in The House Of Lords, though, would the rules of evidence, the ways of how a trial works, etc. be different than that of a trial in a "common" court?

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Didn't they have fingerprint identification in those days? Louis didn't touch the weapon, did he?
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The implication I got was that Lionel killed himself with the weapon Louis used to defend himself in the fight. So Louis' fingerprints WOULD have been on the weapon.

The fight is part of the irony. Normally somebody in Louis's position would hae reported it. But Louis is so accustomed to violence that he keeps it to himself. Thus, he is unable to explain about the fight and the fingerprints at the trial.