There is no app on the iPad, even Adobe Reader, that will handle Dynamic PDF files. (to the best of my knowledge). From your description I'm pretty sure you are dealing with Dynamic PDFs.
But, if you are only talking about editable forms:
Adobe Reader, PDF Expert, and PDFPen will fill out forms with editable fields. That is limited to text boxes, check boxes, radio buttons and the like. They will also let you sign the form, more or less.
Adobe Reader, while free, is the most limited. It can edit the files but has no file management. You should be able to open the files directly from DropBox, or use a file manager app like GoodReader or iFiles, then use Open In or email to output the documents. Do not confuse the PC version of Adobe Reader with the much more limited iOS version.
PDF Expert is a mature PDF app with the most features. It has excellent file management and can access DropBox directly.
PDFPen has nearly as many features as PDF Expert and a more user friendly interface, but is still not as stable (crash proof) as the other two. (according to reviews I've read)