In regards to your questions:

1.  Re Gap Filling:  Because there are various types of gaps, the question can not be answered simply.  I suggest you check out the blog archives for a post of mine made on 12/01/07 which was a fairly extended account of the different types of gaps and their trading implications.  Somebody copied it to the current blog a few weeks ago, but I don't know what date it was on.

2.  Re  Negative Divergences:  As you apparently understand, a negative divergence occurs when a trading instrument makes a new price high but an associated momentum indicator does not make a new high in conjunction with it.  If the trading instrument goes on to make another new high, it does not necessarily negate the negative divergence.  Often a third, or even more, price highs are made that are accompanied by lower momentum indicator highs in which case the negative divergence is still in place (successive negative divergences).  If the momentum indicator makes a new high though, then the negative divergence is negated.  However, such a situation may be a divergence trap.  The best treatment of divergences that I've ever read is in Martin Pring on Market Momentumpp. 9-25.  Here is what he wrote about the divergence trap:

" The Divergence Trap:  Most of the time divergences proceed in a fairly orderly way.  By this I mean that they get progressively lower or higher depending on the direction of the trend.  Then, just as you expect the price to drop..., a final rally develops out of the blue.  Normally, this advance will push the momentum indicator back above at least one of the two previous peaks, causing the wary trader to surrender his bearish sentiment.  Typically, this latest rally will prove to be a 'divergence trap' after which the price will thenfall in the manner previously expected.  This final burst will probably result from some unanticipated news event that causes short covering.  When the short-covering ends, there is very little to support the price and down it goes." ibed. 20-21.

3.  Re Stochastic Crossovers:  A %K line crossover of the %D line is one method of generating a stochastics buy or sell signal, depending on whether the crossover is upward or downward.  Not all crossovers are of equal significance however.  Upward crossovers are much more significant if occuring from an oversold condition, i.e. below 20, and downward crossovers are much more significant if occuring from an overbought condition, i.e. over 80.  Most technicians don't consider a sell or buy signal to unless the crossovers occur in an overbought or oversold condition.  Some technicians require not only that the crossover occur in an overbought or oversold condition, but that the %K line also move above 20 for a buy signal or below 80 for a sell signal.  It is also more significant if the %K line makes a right-sided or through the bottom of the %D line rather than from the left side.  In the former case, that means that the two lines turn up or down simultaneously while in the latter case the crossover occurs when the %K turns up or down and crosses over before the %D line turns up or down.

I think you would be much better off asking academic questions such as these after trading hours as they would be much less likely to get buried and forgotten about, as traders are focused on their trading during the day and therefore much less likely to respond then.