I shared describes in details what are the possible reasons for the compilation errors you have faced. The case on your end, is that the missing elements from the compilation error are only referred in coded steps. This is how these are no longer associated with any step or test and thus are not copied into the new project.
The reason why the compilation error didn't appear in the initial project, is that the elements exist in the repository and can be used although they are associated with other steps.
To solve the errors and be able to execute the test from the new project, you will need to add the elements manually in the Elements repository. The easiest way is to execute the test to a certain step using Run->To Here from the step context menu
and once the recorder is attached to the browser, highlight
each of the missing elements and add these to the repository. That way the element will be available and can be used in the coded steps as well.
In case this solution is not convenient enough for you, you could share your initial requirements and intents why you need to separate that single test in a new project.