In the simplest sense a digital signature is a special checksum of all the bytes in the bytes in the PDF combined with all the bytes in the digital signature. If the document is changed in any way the checksum changes, so the signature becomes invalid. A valid signature ensures that the document hasn't changed since the signature was applied. That stuff that's shown on the signature field is irrelevant. It's just a pretty picture. The real stuff is going on inside the PDF where you can't see it.
Obviously it gets a lot more complext then this. The signature can be selectively applied so that the certain types of changes are allowed. A signature does not add real protection to a PDF, but Acrobat plays along with this scheme by trying to not allow changes that would invalidate the signature, and logging all modifications to the PDF.
1) There are ways to maintain the visual appearance of the signature on the page. If the permissions have been set to allow it the PDF can be flattened, which also partially removes much of the stuff that's going on internally. But this isn't always possible. I had to solve this problem once for a client, so I wrote a plug-in that strips the signature permissions from the PDF (I'll post this to www.pdfscripting.com sometime soon), after which the the PDF can be flattened, keeping the appearance. But of course, after this point the document has to be considered invalid. For example, if the document was a contract, the flattened version would be good for use in an analysis of say all contracts for a particular year, or for copying legal language to another contract, but it couldn't be used as the basis of a legal dispute.
2) I don't understand this one. Do you mean you're deleting the signature field and some of the internal digital signature stuff is still hanging around causing problems? If you own the signature (or the permissions allow), then a standard form reset will "Un-Sign" the PDF. Inserting PDF pages into another PDF will strip out all this info as well.
3) The stamp appearance is set at the time the stamp is applied, it cannot be changed after this point. It's a lot like a PDF stamp.
Thom Parker
The source for PDF Scripting Info
pdfscripting.com
The Acrobat JavaScript Reference, Use it Early and Often
http://www.adobe.com/devnet/acrobat/javascript.html
Then most important JavaScript Development tool in Acrobat
The Console Window (Video tutorial)
The Console Window(article)