In the battle at the beginning we see Squall wound bleed twice(!) and we see the ground and blood on it..
As sparkie said, that's not weird at all. Have you ever cut yourself? Considering the manner in which Squall was cut, it is perfectly reasonable that there'd be an initial splatter of blood induced from the cut followed by dripping blood from the wound. Nothing strange at all.

And since i think strange to see someone bleed twice like that and show it to us... and i wanted to find something so i made a research on this symbol(i knew i had saw it somewhere) and here's what i found...

http://www.symbols.com/encyclopedia/02/0213.html

A clue, perhaps...
It's just a spillage of blood. The sequence lasts for what, 1 and a half seconds, if even that? And you're saying that this tiny little snippet of blood is some sort of deep clue into FF8's "true" story? No, I'm afraid it's just random. You're overanalysing to an incredibly high degree.

and an other discovery... perhaps once again..
http://www.symbols.com/encyclopedia/49/4913.html
That would be something if it wasn't for the fact that while the previous symbol actually resembled the blood in a discernible manner, this symbol doesn't look like it at all. The only thing linking this symbol to Squall is the date, but there are hundreds, even thousands of things that are related to that date, so this is irrelevant.