Carrier planes, as a rule, are far more stout than a land based aircraft. Naval aviation had arrestor hooks, that could be lowered and catch one of the six arresting gear wires that were stretched across the deck. These were steel cables. If a naval plane is heavily damaged, the controlled crash that was a carrier deck landing might be to much for the airframe to handle. These planes might break up. In those cases, the damaged plane might be taken down to the hanger deck for repairs, but it also might just be shoved over the side of the ship and essentially thrown away. It really depends on how damaged the aircraft is, or if they had the spare parts on hand to fix the plane. In 1942, the US Navy would tend to keep and repair their damaged planes. By 1944, the US Navy had so many aircraft and carriers, they were more likely to just chuck any damaged plane over the side. Again, this depends on the damage. A propeller could be easily replaced. Fixing a cracked fuselage or wing that broke off the airframe was another matter. Those planes were far more likely to be jettisoned.
Source: The History of US Naval Operations in World War II by Samuel Elliot Morrison. Specifically the two volumes that covered the battles of Midway and the Philippines Sea.

An airplane lands on an aircraft carrier ship, is held by a rope/spring type of thing and then almost makes a front flip and hits the deck very hard when landing. By doing so the wings smash against the deck and catch a lot of damage, sometimes even breaking.

The aircraft barricade is used to stop aircraft independent of the aircraft's arresting hook.

This was used a lot in World War II - in part because damaged aircraft may not have the ability to lower its hook or may have had engine issues preventing it from landing in a normal profile - but more importantly also because angled carrier decks did not exist yet. As thus, aircraft either landed properly or crashed straight forward into all the parked aircraft on the front of the carrier.

There were a lot of different footage where the aircraft caught damage when landing, what's up with that? Did they repair those things right on the deck in a short time or did they have to go back to the US?

Below the flight deck, the carriers had hangars that could repair the aircraft and machine shops to produce necessary parts. So the simple answer is that simple/easy fixes could see aircraft patched up quickly if the damage was minor.

However, if the aircraft was a total loss or there simply wasn't room on board for a damaged carrier, they could just toss it over the side. The Navy received over 30,000+ aircraft during WW2 alone so replacing aircraft wasn't an issue