Removing Heat Sheild and Misc Exhaust Part

Hello!

Instead of a TL;DR, I highlighted below the two specific questions I have.



I own a 2010 Prius, and have been having rattling issues since a few months after I bought it. (March 2019) Recently, the rattle got significantly worse, and after looking under the car, it was clear the heat shield was loose.

As I was unable to lift the car myself, I brought it into a shop to see what it would cost to re-attach. They said they would not be able to re-fit the shield, and instead suggested removing it. They insisted it would not cause any issues, and while I was hesitant, I figure this at least would solve the rattle in the short term.

I had done some research, and it seems the general consensus is to use JB Weld, or large washers, to re-secure the heat shield. I intend to do this, but my first question is:

1) How urgent is repairing a heat sheild? I currently have the heat shield sitting in my trunk, but would have to buy a hydraulic jack to be able to even start looking at re-attaching it. I would much prefer to get through Christmas before even touching the repair, but wanted to ask if anyone has had any issues doing without a heat shield for a few months over the winter.

This brings me to my next concern.

After the shop removed the heat shield, they were driving my car back out of the garage, and could still hear a (different) rattle. This was after I had paid, and so I was just waiting for the keys.They put the car back onto a lift (without asking), and then one of the mechanics came out and gave me a piece they removed (without asking), saying something about how rusty the exhaust is, and that the piece wasn't necessary in the vehicle.

Ive spent a bit of time trying to find where the piece is on the exhaust, and I found a few pictures, but was unable to figure out the name of the part, or how necessary it actually is to the Prius' exhaust system.

Here are some images:
The first two are images I found online, where I circled what I believe is the correct part. The third image is a picture I took of the actual part they removed.

2) Does anyone know what this part is (images above)? Or what role it plays in the exhaust system? Is this something that I can just leave removed, or should I look into repairing/replacing it?

I'm glad they finally fixed the 2+ year rattle, but this was my first time using this shop, so I don't have much trust in them (yet) and I just want to make sure everything is a-ok.

Thank you!