New York City circa 1905. The Elevated, Eighth Avenue and W. 110th Street.

New York City circa 1905. The Elevated, Eighth Avenue and W. 110th Street.
Wonderfully detailed picture. That guy coming right at us in the wagon with the white horse has his left (driver's left) wheels in the track. Seems like that could tweak those narrow wheels, I wonder if its intentional. This time the guy up on the building on the right has actual scaffolding, though its not level.
The wheel is wider than the gap where the flange on the tram wheel engages so in fact the cart wheel cannot go into the gap in the track. Pushbike of the day on the other hand.

The bloke is a window cleaner standing at one end of a long basket suspended on ropes and pulleys attached to the roof. Its lopsided because his weight is at one end. Were he in the middle it would be dead level.
I'm no expert on scaffolding, having only been on some once when I had to go out the third floor window to rescue a cat, so I'll give you that one, although it seems kinda ridiculous to have that tip to one side when the worker walks over there, effectively creating a ramp to fall off of. But I've seen stupider things.
However I am an expert on wheels, having been a motorcycle mechanic for 35 years after I got out of the army in '87. I've rebuilt, re-spoked, rerimmed, trued and balanced more spoke wheels than I can say, mostly dirt bike wheels but a fair amount of street cruisers as well. The wheel's strength is from the rim in line to the hub, it's weakness is side loads. A motorcycle wheel has spokes which triangulate to the rim giving it some side strength though not a ton, but a wagon wheel has only one spoke with no triangulation so "catching an edge" is death of that type of wheel. It doesn't have to actually drop down into the track. My guess is keeping those kind of wheels true was a challenge; just look at the left rear wheel of the first horse cart on our left with the white top. Also note the heavier wagon on the right; the spokes are flat and wider toward the hub to give it some triangulation and side load strength, though that makes for a heavier wheel.
I'm no expert in anything. I just have experience of some things so know what I am looking at sometimes.

The basket can only tip a bit due to the positioing of the ropes and their proximity to the ends of the baskets.

As to the wheel. I used to push a two wheeled wooden handcart around the shipyard carrying tools, pipes and scrap about from job to shop etc. The surface it traveled on was concrete and some tarmac with inlaid train tracks so essentialy the same as the horse and cart in this picture as stone is just like concrete.
The cart had wooden wheels and a steel tyre. Not once did any cart wheel break, not once did it struggle to cope with the weight. Often times the cart was at one end of twenty foot long six inch pipes and I was at the other end pushing and steering. It had all the weight and it never failed to move smoothly. The axle was wood and steel.

Despite all the abuse it got by being used by different people over the years often doing things with it it was not designed to do, not once did anything go out of true. These steel tyredwooden wheels are much stronger and more reliable than they look.

Have a watch of this video wooden wheel, steel rim, solid rubber tyre. Likely the same combination as was on the cart behind the horse in the photo.