[JENKINS-74149][JENKINS-74148] Extract inline JavaScript

[yaroslavafenkin]

https://issues.jenkins.io/browse/JENKINS-74149
https://issues.jenkins.io/browse/JENKINS-74148

There's a regression introduced in d227992. JavaScript essentially was doing nothing since the content of the dropdown was always shown on the screen, so I've brought those styles back.

Testing done

Before the change
After the change

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