@khasaab, breaking down your queries:
This is possible if you collect your data in Online-Only (single submission) or Online-only (once per respondent) mode as outlined in the support article Collecting Data through Web Forms. But the downside of this approach is that it will not let you save your form (if you wish to complete it later).
Hiding a project name is not possible in KoBoToolbox. However, you could hide the grouping icon if you manage it to skip the entire group as outlined in the post discussed previously:
