Undercarriage damage

Good afternoon,

That is a strut for the motor mount.

You need to take it to a shop and have them raise it up in the air and see if it just came undone or if it broke and needs to be attached.

Roy

Well the good news is that the rusted off part isn't likely to be causing the strange feelings in the steering, it is a hanger hook for the exhaust system. In your third picture you can see the remainder where it curves around the pipe.
That said, you need to look at the system to be sure it isn't hitting or rubbing on a tire, it shouldn't be close enough to hit but it depends on how far the engine can flex.
I would suggest you check the engine mounts for damage, they can make the car handle strange as could bad suspension bushings. To look at the mounts you can open the hood and have someone hold the brake down good, then shift the transmission into drive and rev the engine just a bit, then do the same in reverse. Put the transmission back into park. If you have a bad or worn mount the engine will shift around a lot while the engine was revved. Anything over a 1-2 inch flexing is suspect.