Welcome aboard 69, looks like you'll need a big can of brass polish too.

That rod from the steering box forward to the steering arm on the spindle is called a drag link. The one that spans the front, from spindle to spindle, is the tie rod. As for your shimmy, start with the basics. Your comment about the wheels wasn't quite clear to me. You should have the wheel and tire combination checked for balance, truness (does it wobble side to side), and for round (does it "move" up and down in rotation). This can all be done on a computer balancer by a qualified tire person. If those all check out, including checking the tire carcass for broken belts, or other defects, then move on to the hardware. Often on these types of axles, the king pins could be at fault. The king pin is the, roughly, cigar sized shaft that passes through the upper boss of the spindle, through the axle end, and then through the lower boss of the spindle. There are bushings in the bosses that can wear, as well as the king pin itself. The axle end may have a bushing in it as well, or it may have just been machined steel. If it was an "economy" grade of steel, that hole could be out of round and allow the king pin some excessive movement. When I say excessive, I'm not really talking about a large amount, which may be why it doesn't show up until higher speeds/harmonics. If the axle end is worn, it may be machined for a bushing, if there's sufficient material, or you may have to acquire a new one. Also check the tie rod and drag link ends, as well as the steering box, for excessive wear. And then have the front end aligned.

The configuration of your car looks very much like that produced by Total Performanc, www.tbuckets.com . You may want to contact them too as they've been doing this for a long time. Your comment about replacing "the rack" I don't think would apply as a steering rack is not compatible with a drag link actuated steering.