1. Yes, basically.
The strut bar connect and reinforced the frame to prevent body flex (and making the structure of the vehicle stiffer and more solid, which means less flexing of the shell when cornering.
--If a vehicle has a strong shell already, the strut bar might make no notable improvement at all.--
-On SOME vehicles, the whole structure at the top of the struts can move significantly. In those cases, the strut car keeps them locked in place, allowing the suspension to work as designed.
On the xB2, the significance of the strut bar is debatable (hence my thread asking that question).  It isn't as solid as a Miata (where it is much more for looks than any function), but more than many normal cars.

The Sway bar is connected to the moving suspension bits and serves to make the cornering more crisp and the reduce boxy roll.  Basically, it makes the suspension work better.

2. The xB comes with a FRONT sway bar only.  So, the TRD rear one is designed to balance out the front and make the handling more balanced.

The benefits of sways are well spelled out on the Sparks site:
Benefits:
    * Dramatically improves cornering potential due to increased front/rear roll stiffness
    * Offers distinct handling advantage over just stock front sway bar set up
    * Balanced front and rear sway bars allow xB to handle exceptionally well.  Our unique front sway bar offers excellent chassis repose and quicker directional changes
    * Comfortable sport ride quality
    * Easy to install and thoroughly tested for the utmost in durability and performance.
(Taken, wholesale from: http://www.trdsparks.com/displayparts.php?Mdl_dtl=570&Part_cat=11&parts_id=2148)
The Hotchkiss has a larger bar, so it makes a bigger difference than the TRD set.  They also replace both bars, to have a good balance.  The rear hotchkiss with the stock front would make an ill-handling xB.

Also, sway bars are what used to be known as "anti-roll" bars.  the name has more to do with "rolling" or "swaying" in the suspension than the vehicle body.
Springs and struts have a LOT more to do with body roll than sways.

Of course, it has been a decade since I drove a vehicle fast on mountain roads without a good suspension design and set up.  So, I might have that imperfectly remembered.

Now:
3. We can get all three.
The strut bar is sold by TRD in a well designed one-piece configuration, making it stiff and actually adding additional structural rigidity.  The question is whether that extra rigidity is really that relevant.  Some other strut bar designs flex as much or even more than the car does without them.  As such, they are functionally useless and purely for looks.
The rear sway that balances the FACTORY INSTALLED and stock front sway is the TRD one.  For most people, that is plenty.
Full sway set is also available.  That would be both a front and rear bar sold together and is supposed to make a well balanced.  The only need for such is for a well balanced, but more aggressive, set up than the TRD set.  The Hotchkiss seems to be a good one that does what it is supposed to do and makes a well balanced result for those that want more than the TRD will do.

Unless you take mountain twisties "quickly," autocross, or the like, most folks will tell NO difference between the Hotchkiss set and just adding the TRD rear and keeping the stock front bar.

4. What you would best like in actuality depends on how you drive and various other factors.
Also, I'd advise looking into a good set of stiffer springs to reduce body roll.  That does more than a strut bar.  As a strut bar just acts as a brace between the sides of the engine bay, it makes no difference on body roll, just cornering flex from the struts doing what they are supposed to do (adding "precision" to cornering).
If body roll is what you are after eliminating, I'd look at sways and springs.  Even just the TRD rear sway and TRD springs make a huge difference on body roll with few "negatives" added too.


For my own vehicle, I plan to get the Hotchkiss sway set and either the TRD springs or the Hotchkiss ones.  Because of local bad repaving, I'd lean toward the less drop of the TRD springs.
One day, eventually, I MAY get a strut bar.  But that is down my list after sways and springs (done together) and TRD intake (or the TRD blower, maybe).
For comparison, my Miata is set up for good street driving, while having excellent cornering performance.  I use Racing Beat front a rear sways (replacements for the stock front and rear sways the Miata comes with) and Koni shocks (1/4 inch drop all around) on stock springs.  So, it is simply a more fine tuned version of Mazda's designed system.  A friend the builds autocross and racing Miata (inc. Spec Miata) says that my car is the best street set up he has ever driven and is amazingly balances (I asked him to corner balance it and he refused, saying it is so close that it wouldn't be worth my money or his time), oh this guy also does my 30/60,000 miles services too.
Why tell all that?

I am going for something similar for the xB.  I don't want the extreme, but I want an xB with good street characteristics that has improved handling and can drive quick on a mountain twisty.  Also, a good neutral handing balance and predictability of what the suspension will do is more important than actual body will.

For improved handling, my priority is sways and springs, then struts (stock ones are plenty good for now) once I find a set that performs well over a range of usage and works well with the sway/spring set-up I get.
Only lastly, as almost and afterthought comes the strut bar (which I am not totally convinced is even necessary, as the strut tops aren't all that far forward of the firewall structure).

Yes the shocks come in four. they probably just didn't put the rears in the picture when they took it.

Performance, aside from appearance improves but you could only tell if you actually autocross the vehicle or use it on the race track. For daily drivers, there's really no noticeable difference, except the pride in modding the box