(MaryMac15535's mom here) Since your dad can't file his 2012 taxes until the day the CSS Profile is due at USC, he'll have to do the CSS Profile with estimated figures. (I've looked at the worksheet, and it calls for income information from 2 tax years. If your dad has a feel for how last year went for his business compared to previous years, he may be able to "guesstimate" a bit for the 2012 amounts.)
My spouse is self-employed also, so we are rushing to get our 2012 taxes filed this month. Some of the Common Application colleges our daughter has applied to have financial aid application deadlines of February 1, while others have February 15 deadlines. We will try to get everything submitted for all the schools our daughter has applied to by February 1. Although our daughter has also been accepted by 3 colleges which don't require the CSS Profile, we still want to complete the FAFSA by February 1 (as we've done the past 3 years for her older brother), to maximize availability of Federal need-based aid.
Don't have any advice on the non-custodial parent situation, although what sybbie719 suggested sounds reasonable.