Can I have the signed document attached to an email?

Yes, you can configure emails sent on the signing completed event by using merge fields designed to attach a document in the email body.

The merge field to use in an email body to accomplish this is:

[Attachment - Final Document]

for the signed document, and additional values that can be used are

[Attachment - Envelope Completion Report]
[Attachment - Original Document]
[Attachment - Document Completion Report]

See documentation on merge fields here.