As for your check gauges light, it could be a poor ground or bad battery cable/terminal connector that is causing your issue as well. Also your alternator may not actually be "brand new" most of the time when you buy these from an auto parts store they will be "remanufactured". As such if they did a bad job on the reman then your alternator could certainly be the issue.
As far as your shaky steering, check your break pads and rotors to make sure they are not excessively worn this especially if you are having the issue when braking. If you are also experiencing it while driving it could be an indicator of worn or unbalanced tires or worn suspension parts.