Why was yesterday (Monday, 01/02/2012) a day off for most people?

To add to what I said earlier and address some of the stuff in the OP, federal government agencies were off because they have to abide by a 1971 executive order that grants certain holidays as paid days off in addition to weekends. They cannot get by saying we'll pay you for Sunday, even though you'd be off anyway. It's basically part of a contract with federal employees.

Banks are generally off because the Fed is closed and transactions to and from them will not process. It is also usually included in employment contracts.
As for why your stuff didn't process, that would be up to the individual banks. If it was set for the 30th then I could see where you would have a complaint. I worked at a bank for several years in college. I did end of day processing sometimes and we run everything after close everyday. In the current situation we would have run COB for the 30th on Friday night, then also ran end of month (31st processing) right after that. They process everything for Saturday and Sunday on Friday night every week, as well. They also were closed yesterday so they would have processed all Jan. 2nd business that night too. I still bank there and know they did because my wife gets paid on the 1st and it processed about 10:00 Friday night. Some banks choose to not process stuff for Saturday and Sunday until Monday morning. In this case they would have held Sat., Sun. and Mon. until Tues. morning.

For schools, they set their own calendars based on the requirements of students being in class for 180 days (teachers have 187). They can set them how they choose. One district in my area started yesterday. My district had teachers in yesterday and students in today. So I worked yesterday even though the rest of my family was off.