I think this is quite late for me to post a response, but unfortunately, no you cannot get paid for the days you work before the start date of your OPT date. I had a similar situation too, where I got a job offer but my OPT doesn't start until exactly one month later. This has cost me my job offer and I got replaced by someone...
The field I am pursuing is a highly popular and difficult one to break into as well. I cannot understand how the US government can make our live here so difficult when we paid so much money to study in the US already... But obviously the locals will always have the advantage first.
I am not sure how is your situation now but I hope you manage you keep your job.

I'm back at sweating away about landing on a 2nd job offer before my OPT expires.