The Mk1 and Mk2 both had McPherson strut suspensions. The Mk3 and Mk4 have double wishbone suspensions.
http://www.toyotaracing.com/trd/parts_catalog/chasis_suspension/suspension_types.html
Strut suspensions are lighter weight and much cheaper to manufacture, but have larger limitations on camber adjustment and adaptability. They feature only one "A" frame at the bottom, with the shock absorber itself forming the upper mount for the steering knuckle, or for rear wheels, the wheel axle. Because the top mount is to the body, and not a subframe, as with our cars, the top mount point is subjected to stresses from cornering which can cause the top mount point to deflect from it's nominal position. In this case, a strut bar will help reduce or eliminate that deflection, and thus assist in handling characteristics keeping the tires flat on the ground.
The double wishbone front suspension of the A7 and A8 chassis precludes the necessity for such a piece, making it a purely appearance item. In this case, all of the stresses stay within the subframe, and none are transmitted to the shock tower mounts, so the brace does nothing. Well, it COULD do something in the event of a collision, like compromise the car's crumple zones or the like.
Targa flex is real, and measureable. It is mostly in the floor between the doors. The door hinges and latch take up the stresses on the sides, so the interior panels won't show signs. You might in the door sills though.
As to the the difference between a strut and a shock, it's really that difference between suspension types, however, many people call our front shocks "struts" because of the mounting pattern. Because struts have to carry lateral load, they need a more secure mount to the car body, and they typically have a large flange on the top to do that. Bottom mounts are typically 2 or more bolts along a vertical flange. Old school shocks used to have open eyeholes, and a single bolt at top and bottom to mount, allowing the shock to rotate as it needed. This is what we get at the bottom, but we get that typical "strut" mount at the top. Another typical difference is that shocks had nothing to do with the mounting of the springs. Because with a strut, the absorber IS the top mount, the springs had to be mounted to them, and the spring mounts would be on the absorber casing. This is the case for the A7 chassis, which is another reason why many people call it a strut. Most traditional shock absorbers did not have the spring mounts on them