If the ball hits the ground before anyone gains possession of it, then it’s incomplete. Once somebody’s already had possession (which the first guy who caught the forward pass did), that play can’t be incomplete anymore, and if the ball hits the ground it’s just a fumble. Whoever dropped it looks a little worse on their stat sheet (Fumble) and whoever picked it up looks a little better on theirs (Fumble Recovered).
I think every American lad who’s ever loved football has scored on a crazy play like this once, at least in touch football. On my high school’s parking lot, I was often teammates with one fellow who was known for his Crazy Sports Antics. Tyler, being as good a quarterback as he was a receiver, would have rather thrown a lateral than been tackled on any given play. When his brother was on his team, they were unstoppable. Entire defensive strategies had to be invented so that Tyler’s would-be receivers were covered adequately, even after Tyler himself had caught the ball. Of course, when someone else on his team caught a forward pass, he was right behind them to take a well-timed lateral. Ah, those glory days…
BTW, I thought I’d translate some things in the concise clip of “The Play”.
Squib (“he’ll probably try to squib it”) - Normally, when you kick off to the other team, you want to boot the damn thing as far as you can (as long as it doesn’t go in the endzone or out of bounds) to pin the other team back. The major disadvantage of this is that, since your guys can’t advance down the field to tackle him until the moment the ball has been kicked, the kick returner generally gets at least 5-10 yards for free, and since your coverage team will be so spread out, it’s possible for a really good return man to break out completely and score. In a squib kick, you intentionally kick the ball for a short distance, hopefully with enough hangtime that your boys can be breathing down the kick returner’s neck by the time the ball gets there. Doesn’t always work, apparently. 
“The Bears have to get out of bounds” - They apparently had precious little time left in the game to make their comeback. Going out of bounds, throwing an incomplete pass, or changing possession stops the clock automatically. (I think the clock stops on first downs in college football now, too. I don’t really keep up with it.) A kickoff results in a change of possession, so the clock stops as soon as the return itself is over–the guy with the ball gets tackled or runs out of bounds. The traditional strategy for a team losing very late in the game, like Cal in this situation, is to get out of bounds quickly and take bad field position in exchange for more time; then, once the ball is yours, try for a long forward pass to a guy very close to the sideline who can then run out of bounds before anyone gets a chance to tackle him–if they do tackle him, the clock will keep running and you have to resort to other, more difficult tricks. If you’re down by 1 or 2 points (or 3, if you’re willing to go into overtime), you do this for a couple of plays until (ideally) you’re close enough to try a field goal–and there are about eight seconds or less left, so that the game will pretty much end with the field goal attempt. If you have to score a touchdown to win, it’s much harder–unless you’re Tyler or you played for Cal in 1982.
“The 30”/“The 20” - the line that marks 30 yards/20 yards “to go” for a touchdown, respectively.
Flags (“there are flags all over the place”) - Referees throw yellow flags on the field when they see a penalty occur. It doesn’t automatically mean that a penalty has occurred; sometimes they’ll confer and decide that, no, there really wasn’t anything untoward, it just looked like it. Yellow flags inspire fear and loathing in a team that’s just scored; if the penalty was on them, the touchdown doesn’t count and they may lose yardage as well. If the penalty was on the other team, it won’t have any effect; when a penalty happens, usually the play keeps right on going and then the referee will go over to the team that didn’t break the rules and ask if they want to assess the penalty. If the defense were penalized on a touchdown-scoring play, the touchdown would be taken back and the offense would just have the ball a few yards forward of where the penalty took place (generally), so of course they’ll decline to press charges in those situations. I don’t know what the penalty was on this play, if there was one–obviously, it wasn’t against Cal. These days, BTW, head coaches (managers) also have flags–those ones are red, and they throw them when they want to challenge a call with instant replay. The timing of that throw is crucial; a split second too late and he blows his chance to challenge the call.
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