The 3-way appears to be a simple pickup selector, one end or the other for the individual pickups and middle for a straight blend. The middle terminals being connected on this switch is actually quite useless as the two wires from the pickups could easily both be on the same side of the switch, eliminating the need for the crossover. But this way should work fine too, of course.
As for the 2-way, it looks like the middle terminals are connected to ground. If so, then this is nothing more complicated than a simple coil tap. When the switch is flipped, one coil of each pickup is grounded, putting them both into single coil mode. The 3-way should continue to function normally, giving you a total of six possible pickup combinations.
Everything else about the wiring looks very standard. Actually, all of this is pretty normal, except that I've never seen that type of switch used as a pickup selector. I'm sure it works fine, it's just a bit unusual.
Now, I'm not personally familiar with the wire colors on Lawrence pickups, but if
this is to be believed, they're a bit different from DiMarzio's. Matching up what equals what via that chart should allow you to install the new pickups the same way the current ones are, but if you want to do something else with it, I guess you'll have to figure that out first.
