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The "thing" that broke is a splash guard. The "tube" is your exhaust and that "emission" thing is a muffler, which has absolutely nothing to do with emissions. All it does is reduce the amount of noise produced from the exhaust. It's fine.
You should be able to reattach the mud guard. They only mount with a couple of screws into the plastic wheel wheel trim. You might have to make new holes if the original holes can no longer take the screw threads.
The dent in the exhaust pipe might create a bit of restriction of the exhaust flow, but I doubt you would notice it affecting the car's performance. As long as the exhaust pipe is not punctured, you should be fine.
Lesson learned judging curbs. You would not want to pay for a damaged wheel - which could be much more expensive.
The dent in the exhaust pipe might create a bit of restriction of the exhaust flow
Fun fact there was actually a test done to see how dented exhaust would reduce the engine performance, and truth be told the performance actually increased. They beat the hell out of the exhaust and it continued to increase 1-2HP with every dent they caused. Of course eventually after so much denting it did decrease but yeah fun fact!