Is it mandatory to give the BOE a copy (assuming one of them has not signed it)?
Anybody can sign it technically but of course they recommend that it be a witness. After it is signed the Elders are telling people to make three copies: one for their health care agent, one for their doctor, and finally one for the Elder's records. They used to suggest this but most people just ignored it, now they are saying everyone "needs" to get this done.
Excuse my ignorance, but is this DPA just over health matters or does it allow the designee to control other aspects of the incapacitated one's life, like finances?
No, the wording of the document makes it clear that this is for health care/end of life decisions. I believe there are other types of DPAs for financial matters.