Ok, firstly LSD.
As already mentioned the diff (short for differential for anyone who was wondering) essentially allows the drive wheels to spin at different speeds. This comes down to the power being delivered but is a much simpler way of thinking about it.
A limited slip diff (LSD) simply limits how much the wheels are allowed to spin at differing speeds eg it keeps the speed of the wheels in question closer. This makes sliding around easier to achieve and control.
A welded diff is one that can't slip at all, the wheels must always turn at the same speed. Welding is a very cheap and nasty way off accomplishing this. The end result is something like when a 4WD has the diff locked. It makes going in a straight line or driving slowly over big rocks easier however this will not make cornering very fun. Think about rolling a tube of cardboard along the floor then turning a corner with it - to do this you have to make one end turn faster than the other, otherwise it doesn't work very well.
As far as what cars have LSDs and what don't basically 'sports' cars often come with them (eg all tickford vehicles did). They are a common mod especially on the racing/drifting scene.
They are more common on RWDs because RWDs are the dominant type in the sports/race/drifting arena. Also installing one on a RWD is quite easy, usually take an afternoons work. Unbolt existing drop it out bolt in LSD clean up and you're done, just have to be careful not to damage anything.
Whereas to install one on a FWD you will usually need to remove the transmission(gearbox). This is because in a RWD it pretty much goes sequentially engine -> gearbox -> drive shaft -> diff -> rear axles -> wheels. All nice and seperate. Whereas a FWD goes engine -> transaxle -> front axles -> wheels. The transaxle entailing both the transmission and diff. So basically it's more of a hassle to get to, more of a hassle to do and compatibility issues make aftermarket LSDs harder to come by.
Converting AWD to RWD:
This pretty much depends on how the AWD is managed. eg whether it is mechanical or in some way electrical. The Adventra for instance is mechanical using a planetary centre diff (much like a 'normal' diff which controls how much power each wheel gets a centre diff controls how much power each end of the car gets eg an AWD has three diffs, front, centre and rear). Because it is mechanical like this the only way to adjust the torque split (fancy term to describe splitting the power) is to replace the diff.
On the other hand a skyline can be converted to RWD by popping a fuse, because of this for many years there have been aftermarket piggy back units that plug in and allow the driver to control the torque split in real time from the cabin. The Sti WRX has just introduced this feature as a stock feature and the MPS Mazda 6 uses similar technology to facilitate drift. I have no idea how it works really but as a heads up its out there. Converting AWD->RWD can be either tricky or easy....
Strut Braces:
This has been explained pretty well so I won't go into too much detail.
Basically when cornering a cars suspension is flexing everywhere to absorb the inertia that wants the car to keep going straight. As a result the car will most likely tip foreward towards the outside front wheel.
A strut brace or any number of other suspension components (stiffer shocks, stiffer/lower springs, coilovers, sway bars, polyurethane bushes, lower profile tyres, bigger wheels, etc, etc) all serve to stiffen the car. That is to stop the car flexing as much. To stop it absorbing as much interia. As a result the car is much more predictable and thus easy to control when cornering. The variables are reduced from every suspension component to steering and speed which are controlled by the driver.
The other aspect is that a stiffer body absorbing less inertia will exhibit more inertia. It will want to go in a straight line more, the immediate result of which being the car will like to slide more. This is great news if you like drifting.....
Reasons for strut braces: An engine swap will often introduce a more powerful engine into a car not designed to handle it. The engine alone exerts all kinds of pressures on a car and to help resist these a strut brace may be used. In fact to get an engineers certificate approving the swap, one will often be insisted upon.
Finally the popularity of strut braces: They are very easy to install, they make the car slide and they are easy to show off.