Point 1:
Your CURRENT employer (not the former one) almost certainly has to provide you with a copy of the reference from your previous employer if you ask for it. See "Do you have to give a copy of a reference you have received from someone else?", in this document (explaining the provisions of the Data Protection Act) from the Information Commissioner's Office:
http://www.ico.gov.uk...rences_v1.0_final.pdf
Since, for the first year of employment, any employer can dismiss any employee without the need to give any reason, Jobcentre Plus (and the DWP) must be used to people leaving jobs after only a short time, without any loss of JSA. You'll probably find that your claim is unaffected by your circumstances.
Chris