Google Sheet: Can I re-integrate a form with the existing Sheet?

If we remove and re integrate, can we do so on the same Google Sheet as it was previously integrated? We have scripts and formulas that run on the submissions tab that take time to set up, and can't really do that every time we want to add a field...

Unfortunately, that is not possible. If you remove the integration and re-integrate your form with the Google Sheet a new Sheet will be created. The new Sheet will have all previous submissions. 

Please note that any edits made inside of the Sheet that is integrated with a form can break the integration and we do not recommend it.

If you want to edit your Sheet, you may import the sheet data to another Sheet using the importrange function. This way you can edit the other sheet and avoid breaking the integration.