If you saw the hospital and not the clinic then no, the clinic can not bill you. A healthcare provider has a right to ask for relevant medical history and there should never be a charge for that (different if you're asking for it). In this instance, only the hospital has the right to bill you (or Medicare).

Out of curiosity, was the Dr who put through the claim on duty at the hospital? Either in person or in an on-call manner?

You won't ever be asked to sign a medicare form in hospital - although I think you sign something on admission and they use that for everything. But hospital billing is different to a clinic.

In a clinic setting you should be asked to sign for it before you leave but if you don't (for whatever reason) they can sign it on your behalf (citing the reason). We did this often when the system crashed and we had to do retrospective claims.