ok, i have a few minutes, i will give you a good answer to some of your original questions:
it may be the kite, different kites perform better or worse in low wind situations, weight is one factor, overall design is another, and the tuning of the kite and how you fly it is a final variable.
the kite design is what it is. the tuning and your handling is the variable.
tuning: it is common for kites (especially pre 2012) to be factory tuned for a few inches of oversheeting. it is also common for many brands to recommend a tighter outside line connection point for lightwind flying (more power). for many people, this creates a problem, and in fact hinders light wind performance.
oversheeting: a kite is oversheeted (and will stall - fall backward with a tight canopy and 0 power) when there is too much overall backline tension for the moment.
interestingly, kites can handle more overall backline tension without stalling when the wind is stronger and there is more wind energy pushing through the canopy. in lighter winds a kite cannot handle the same amount of backline tension as in stronger wind. in light wind you may find it easier to keep your kite flying, and particularly climbing with speed, if you depower it and take some of the tension out of the backlines.
other factors that contribute to stalling: the position of the kite in the wind window, and the direction the kite is trying to fly (this is angle of attack) ... upwards is the most difficult angle for a kite to fly (high angle of attack, and fighting gravity). many people know that a kite pointed upwards will climb faster when sheeted out (less backline tension). it is recommended that you tune your kite to the conditions, so that you can hold your bar all the way in and steer aggressively without accidentally oversheeting and stalling the kite.
so .... you might get better lightwind performance out of your kite by depowering it, either by lengthening the backlines at the attachment point on the kite, or by using the trim on the bar.
kite flying: in light winds the center of the powerzone is where you need to be, and the edges of the wind are a problem. tight radius turns and "oversteering" on up and downward powerstrokes (cutting the kite slightly back towards the opposite side of the wind, esp on the way up), will help keep the kite in the wind. parked kites along the edge or above your head are in a weak position. there is a small catch, although tight turns are generally much more effective than sweeping long radius turns because tight turns keep the kite more centered in the wind window, some kites are more susceptible to twisting and stalling when turned tightly. (you can rip some kites through tighter radius turns with more consistent power delivery and flight than others) thats just getting to know your kite.
this is a topic that comes up a lot in lessons, it may not be the answers to your specific situation, but if stalling is the type of "falling" your talking about, then this might help. loosen up your backlines, a lot.