Yeah, check if the headset is loose first. But I wouldn't ignore flex in the fork. Some flex is normal, but hard to describe how much. If it is a bonded aluminum fork, there is the rare possibility that one of the joints is coming loose which is extremely dangerous. I'd have someone knowledgable take a look.
Shudder under braking can be caused by a lot of things. Towing in the brake shoes is normally done to remove squeel, not shudder. First checks are to clean/sand the brake shoes, clean the rim. The clicking could be the seam where the rim is joined going through the brakes (this will normally be opposite the valve stem.) Check for loose parts/play on the brake caliper as well.