For your steering problem is known as brake judder, basically your rotors are warped and/or glazed. If you really wanted to check on it, you would need to perform a lateral run out and it should not be .002" or greater on difference on sides. I would replace the rotors and pads to start off with and ensure when you get them to run a quick sandpaper down to prevent the shipping glaze that's on it.
For the right side sound by the wheel it can be wheel bearings giving out or missing a few. Could also be the pads are skipping the rotor as well. Hope this helps.
2009 Cobalt LS XFE U74
Mods: K&N Typhoon Intake, Option Cat-Back Exhaust, Cosmo Short shifter, Maxxim Ahead 17X7's, MPx shorty antenna, FE5 Strut/shocks, Tein S Techs, Moog endlinks, Moog spring seaters, ZZP shorty header, LSJ downpipe, LNF front grille