There will be others with better/more detailed info....
The LR3 and LR4 are basically the same truck. The LR4 is an update of the LR3. They share the same suspension type and engineering, even if some of the details changed.
The LR3 has a 4.4liter jaguar based V8 @ 300 HP while the LR4 has it bumped to 5.0 liters and 400 HP. There are V6 LR3s, but they don't seem to be common. They use a Ford SOHC 4.0 V6 so that might be an advantage if you ever need engine internals (most externals won't swap with an Explorer)
All have a computer controlled locking center differential. AWD is the normal mode and the computer can lock to 'traditional' 4WD. It's considered very effective.
All have an OPTIONAL HD package that includes a computer controlled locking rear diff. You cannot manually select the locker. It is also considered very effective.
All have a very good traction control system that uses the brakes to simulate a locker when needed. Many folks say it makes the real locker redundant, I say drive both first or just find one with the rear locker. Otherwise the traction control is generally considered the best in the industry.
The LR3 interior is more utilitarian, which some prefer for an overlander. The LR4 interior has softer textures and is considered more luxurious.
The V8 LR3 will accept 18" wheels and tires. The V6 models will accept some 17" wheels, but there is a good selection of 18" tires so most don't follow that path. Because of larger brakes, the LR4 will only accept VERY limited $election$ of 18" wheels, so your usually stuck with OEM 19" wheels that have a crap selection of tires. A few have tried wheel spacers and 18" wheels, but there is no long term data yet.
Both LR3 and LR4 have adjustable height air suspension and 4 wheel independent suspension.
Both are VERY heavy, but strong.
HSE is a trim level, the highest. The trim level doesn't have anything to do with any off-road type of options. The rear locker can be had with any trim.
32" seems to be the limit on a stock truck.
Lifts can be accomplished with either a software tool or modified linkages to the height sensors. Essentially you pump the airbags up. There are no 'hardware' lifts like most trucks. It does have a separate frame, but I've never heard of a body lift being attempted. I'm guessing that the frame and body are tightly integrated.
There is a surprising amount of protection related aftermarket available. F/R bumpers, winch mounts, sliders, skids.