in Try and Catch Me I will never understand why she decided to tell Columbo that she found Edmonds keys. What was the purpose? She was going to toss them into the ocean why not just throw them back in her purse or pocket. Why does she have to admit the Columbo that she had the keys? That one has always stumped me

At any rate, the way it's filmed has always bothered me a little. If Columbo were "right on top of her" when she takes out the keys, it would make more sense to give up and show them to him. But the way it's filmed, he's several feet behind her when she takes them out.

While I don’t disagree, she was flustered when surprised by his voice and may have been worried that he could have seen the keys. The tension continued for her because they were right there in her pocket. Would he say something? Then, when Columbo says that when he finds the keys he will find the murderer, this gave her the chance to come clean (if he saw) with the keys and try to separate herself from his conclusory statement.

What bothers me about this episode is the light bulb. After the murder, and all the police going in and out, etc., didn’t someone ask, “what’s with no light in here?” Nobody tried to change the bulb?

I agree with the analysis above. Miss Abigail was almost caught red handed trying to throw the keys into the sea. Wouldn't that leave anyone flustered and unsettled. She knew somethng Columbo didn't and he was right there unexpectedly. So she made a BIG mistake by saying she had found the keys. It merely confirmed Columbo's suspicions and affirmed his theory she was the murderer. Perhaps she ought to have walked along the same beach in the first scene and buried them in the sand!!!! Discretly of course and miles away from Columbo!

This seems to be an occurring theme throughout the series, the killer helping Columbo with a clue he needs only to hang themselves in the process. Hayden Danzinger and the gloves is another good example.