All the examples of “take a crack at” are correct. The first one sounds the most natural in context.

Great! So, one little note. To “get a crack at” is to be given the opportunity. So, you already have the opportunity to record the video (just follow this link). What you mean here is that one day you will attempt it. In that case, you would say “One of these days I’m going to take a crack at recording a celebration video.”

So you “get a crack at” when someone gives you the chance to try something.
You “take a crack at” when you already have the chance and you decide to take the opportunity.

Yes, exactly – but you need a verb here – “…so he asked his employees to take a crack at the problem.”

This is a good time to use “take another crack at”. Bankrupt is a strange word. When used in the active voice, believe it or not, the subject is the cause of the bankruptcy. So you would say, “The drought bankrupted many farmers” or “COVID-19 bankrupted many restaurants.” When you’re talking about people or businesses, you say “go bankrupt.” For example: “The restaurant went bankrupt after the pandemic.” So in this case, you can say, “The company owners decided to take another crack at business by operating in a brand new field after their first company went bankrupt.”

Great example! Just a few edits: “After a few months away from Plain English, I decided to have another crack at Jeff and JR’s work.”