It's probably clear from the start that the designers had no "actual" functionality in mind when drawing these buttons. Even the ones with meaningful labels on them were probably labeled more or less at random, albeit with buzzwords that as such make sense.
However, it would be easy to argue that the crossed-scrollbars control mechanism places certain requirements on a keypad like this. The mechanism completely ties down one hand in a position that is difficult to maintain: if constant steering is required, at least one finger must be precisely positioned in one location and moved as required. Other fingers will find it difficult to achieve anything simultaneously - unless key functions are associated with buttons within easy reach of the crossed-scrollbars section. And that basically means in the four sectors falling in between the bars.
There's no point in having labels on those buttons, since the user's hand will be covering them and preventing him from reading the labels, so for once the use of meaningless strings of numbers in the graphics makes sense! Most probably, the sector buttons would be user-definable shortcuts to functions whose actions are visible outside the area covered by the hand - say, the keys numbered 15 and 78 would "expand" to the cluster of function buttons visible to the upper left, and these in turn could be used to redefine the function of the keys 15 and 78. Any other sector buttons would similarly expand to other function clusters as needed.
The big "FIRE" and "ABORT" buttons seem stupidly placed, as a right-handed person would be somewhat blocking his left hand access to them if simultaneously operating the cross-scrollbars. Perhaps they are mere indicator lights, and the trigger is actually slaved to that tiny square segment of the upper vertical scrollbar?
Anything to the left of the column that is connected by a curve to the "ABORT" square is probably unrelated to weapons firing, as Worf's duties include other stuff as well. The next column to the left has various directories and ship's departments listed, and is probably used to remotely summon these resources (so that Worf himself merely sends requests and gets answers by people or expert programs at the other end, and no fine control interface is required). The next column probably has a target sight on top, status indicators relating thereto next, and a way to reconfigure the entire display at the bottom.
Constant reconfiguring is probably the very point of interfaces like this. The layout is so distressingly non-functional and non-optimal because it represents the current user's current reconfiguring of the necessary elements for the task at hands, with irrelevant elements left as is. A few moments later, Worf might call up a completely different view, or his replacement officer would choose her preferred layout, or a communications interface would pop up to allow Worf to answer an incoming hail.
1) No logic to it other than placing quick-select, reconfigurable buttons in the sectors between the scrollbars, for the needs of the day.
2) Worf has chosen to redefine keys 15 and 78 (his options including AUX BANK, PROBE SELECT and others), so his previous work redefining keys to the upper right (selecting between torpedo warhead type, apparently) is now forgotten and fades to the background.
3) Nope. But row upon row of identical-looking buttons probably needs a scheme of (randomly) alternating colors so that two adjacent keys aren't confused with each other.