If there is movement in the handlebar, and not just in the brakes. Then check the steering bearings. If those bearings have worn out, then it will shudder and creak when you brake hard. This will also be noticeable if you hit a big pothole or bump. Basically sudden deceleration at the front end. Is this happening when you hit big potholes?
Can you convert it to rim brakes?
It depends. Does the rim have brake tracks for the rubber pads of the brakes to clamp upon? Does the frame have mounts for the rim brakes to be mounted? It might work out to be expensive to convert this as well!
Do rim brakes have less performance?
For commuting and the kind of riding you described. There will be negligible difference in actual real world usage. And life becomes much simpler in terms of maintenance. But that depends on whether it is possible to convert it to rim brakes on your bicycle!