The board should still function fine without a battery. If it doesn't boot up after the battery has been removed or properly replaced, then something else is going on.
The battery solder point looks pretty well corroded, and its harder to tell with the others as they appear to be covered in dirt/funk. I'd recommend cleaning the gunk off of the board so you can inspect it for corrosion and problems easier.
Well I got off most of the glue (this stuff is insane), and it looks like the area around the corrosion was just about ready to flake off because I barely touched with with a QTip and alcohol and well, you can see that a bit of the green layer flaked around it.
I don't think they point is bad, when I plug my battery in (soldered to the back of the board) I can still read voltages from the point, and I can still read voltage on the 4990 chip and the clock crystal. It doesn't look like there are any damaged traces around the corrosion.
Here's a few pictures: http://imgur.com/a/bYePA
Are there any other areas of the board I should be looking at? I should be getting unibios on Saturday, and I've read that it can bypass the error if there isn't a larger hardware problem on the board.