The pictures you posted appear to focus on the cv boots/joints, and it looks like one side is flinging grease. Its possible the clamp may have come loose. This will have no affect on the handling behavior at this point. It will eventually sling all the grease out, and the cv joint will fail as a result of no lubrication. This could turn in to a vibration, and eventually break if not addressed in a timely manner. If it breaks, it just means power is not going to the wheel, which just means no go.

None of this again will cause the issue you are describing. I would be checking the suspension for loose components, and confirm the subframe is bolted properly. If everything checks out, I would take it in for a 4 wheel alignment, and make sure the shop knows what'* going on. If there is an issue, that should flush it out.

Yes, definitely.

I agree with rjolly87 above.

I also find myself with some questions, mostly for diagnosis and partly for curiosity.

It takes a lot to break a sway bar. I've personally never seen a broken sway bar that wasn't on a badly wrecked car or on a web site showing results of a badly wrecked car or someone likely saying "wooooo" right before the break. What kind of hill was this?

Was there some other factor (sudden drop-off, pothole, motocross sign nearby) with this hill that caused so much damage?

How fast were you going on this hill?

Did the sway bar end links break?

By a shop?

Did the "replacer" do any other diagnostics or have any other comments?

Sounds like suspension travel is unevenly hindered somehow.

Sway bars don't really have stress from hills or bumps that equally affect both attached wheels. Sway bars do experience stress from cornering, uneven road geometry (bump on one side, pothole on one side, touchdown on one side first). You can jump a car (minor jumps, not Dukes of Hazzard) and not hurt the sway bar at all if the car lands evenly from left to right. Might bend up the rest of it, but the sway bar would be fine. This makes me think we need to know more about the original damage and events leading up to the damage. This could tell us a lot about what else is going on.

Maybe get us some more pictures too, like of the undercarriage and corner shots of the exterior of the car.

. . . to turn the steering wheel and not have an effect on the direction of the car? . . . or to actually steer the car?

It could be a symptom of the issue.

This seems important. It sounds like it is seriously bad at this time, however, if you are unsure if it was this bad prior to the sway bar being replaced then we might have a whole other issue to diagnose. Did you drive it much between "the event" and the sway bar replacement?

"Sway" is essentially inertially-induced leaning, usually a result of steering input combined with vehicle speed and loading. As these cars aren't known for wind-induced sway and you've mentioned nothing meteorological I am intentionally ignoring wind-induced sway. By "sway" are you steering first and having this result? . . . or are you hitting a bump/dip/etc. in the road and this behavior is induced?

"glued on" refers to the CV boot with grease coming out?

Ah, this might answer my question before last.

Is there body damage on this car?

I've seen this happen on trucks with lift kits that don't take CV geometry into account, and on suspension that'* been overextended quickly with wheels turning, such as during a jump. Usually on trucks with IFS also, but the designs are similar.

It'* possible, however, it wouldn't be likely to cause the issues you describe. Sway bars pivot relatively easily in their bushings. Their job is to be a spring loaded bar that takes excess spring capacity on one side of the car and use it to assist with excess downforce on the other side of the car. If one end of the sway bar was disconnected and you go over a bump that is equal on both sides you'd likely now feel a difference. The difference would come in with cornering, severe crosswinds, and the vehicle being loaded with excessive weight on one side or the other.