That field is not available on the UI, unless you store it in a custom field using some scripting or development or maybe using worklow post functions.
Issue Key is also unique as long as you don't move it to a different project. Even if you move, you can still search using the old issue key.
What script? The issue Key is right on screen in front of you whatever you're doing with an issue, and it's in the url if you're just browsing issues, and in the grid if you're looking at the results of a search, and and and.
You'll need to explain what you're trying to do before we can help you much. Saying "create my script to access it" tells us nothing about what you're doing. (Also, you mention SFDC, which is utterly meaningless without a context)
Um, the script goes into Salesforce, which I'm not that familiar with. Your script should talk to Jira, probably via REST.
You need to explain and/or think through that "synchronisation" you mention - how does that work? If the case is created in Salesforce, then it's that that needs to work out what it's just done and get back the Jira response.
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