Is there an API or a dataset of all US administrative forms? I tried many searches without great success
Without looking into this at all, I'm going to say that no, there is no comprehensive dataset/API of all US administrative forms. That said, you may find help/references/resources by contacting these groups: Center for Civic Design, Civic by Design, the people behind Civic Design Camp, and the Lab @ DC, which very recently held "Form-a-palooza", an event for designing gov forms.
Again, this answer may very well be wrong; if so, I'm more than happy to delete it.
Also, if I am right and this dataset does not exist, perhaps you could start it? Creating a repository on GitHub and adding the forms/documenting their origins is where I would start.