First point. A Public Trust is a snapshot of you at that time. You are “cleared” to work that particular job/task/position but a Public Trust suitability determination is not generally carried over to any other agency/position. To answer the question, yes, you can be found suitable for a Public Trust position and have a security clearance for another. This is common if you work for a contractor that has several contracts with the federal government.
Yes, you will likely be required to submit a new SF 86, as a minimum, if you have not worked in the cleared position for over two years. Even with CE/CV, you can be required to submit a new security questionnaire at anytime, but no more than five years since your last submission.