are you able to solve those puzzles that give you some shape and ask to identify which other shape represents the original after some rotation or transformation
No. Those are very tough. I solve them by naming the shape and then manipulating the sounds of the names of the shape. Like "a triangle pointed upward" rotated 180 degrees is "a triangle pointed downward". But if the shape is hard to name, I have a great difficulty.
The other test that comes to mind is the one where you have some flat two dimensional shape and are asked which ones when folded along the edges will become boxes (or some other 3 dimensional shape).
Again, something that is nearly impossible for me to do.
but it seems to me remembering someone's face would be impossible without some form of visual recall.
I mostly remember people by what they wear, or if they have name-able attributes (like "long blonde hair" or "glasses"). I pick myself out of a crowd because I usually wear the same things.
Learning a dance move is also difficult, because I can't go directly from seeing some move to moving that move. I have to see it, name every piece of it (inventing names as I go along if needed), then repeating the names in my head to trigger my body. And if it starts getting to be above the "7 +/- 2" threshold for total of "names of step" times "steps in the sequence", I can't keep all the different names straight, so I just go random.
For example, every beat of the macarena got translated into some phrase ("right hand on hip", "left hand on hip"), and those movements of that simple sequence are just about overload for me.
IIRC, recognizing people's faces is a very specialized brain activity, which seems to be only partially related to other visual skills or memory - there is even a disorder called prosopagnosia that specifically impacts the ability to recognize faces.
I have no trouble recognizing peoples faces, but I usually find it very hard to remember their name. Also I haven't been able to learn a single dance in my life so I usually just bounce around :-)
update: I usually don't think of anything at all when I count, but when I'm distracted I do use my fingers.
I'm aware of that condition, but I don't have that. I have a more general problem of not recalling something if I don't also name it. That is, unless I tell myself (in words) that you're wearing a blue shirt, I cannot recall that you had a blue shirt.