Which non-rice grains cook up most like sticky rice?

From a purely sticking perspective, I would say golden millet. When I leave a cooked batch in the fridge, it goes to a single slab within a few hours. However, tastewise, it is much more reminiscent of wheat than of rice. It is very mealy in texture, too.

Alternatively, you might want to experiment with adding tiny amounts of starch to a different grain, to make it stick, or maybe using pap instead of rice, possibly embedding some other grains (or maybe something like tapioca pearls) to have some variety in the texture.

Honestly I think you'll struggle on this one. Sushi rice gets its stickiness from amylopectin. Starches, generally, have some amylose and some amylopectin, but certain varieties will have different proportions. "Waxy" potatoes, waxy corn, and sticky rice are all much higher in amylopectin than other varieties, which give them their distinctive texture.

I think @rumtscho's suggestion is good, but simply adding extra starch really won't replicate the texture of sticky rice.

I tried to find information related to the amylopectin content or other cereal grains, but unfortunately found very little that considered the grains comparatively. Bur it seems like tapioca starch generally mimics the behavior of waxy starches, so the tapioca pearls mentioned before might be a good experiment on their own.

In addition to the range of starches you could add (tapioca, corn, potato, arrowroot - all of which will create slightly different results), you can whiz up a few spoonfuls of dry porridge oats in a blender and stir these in late in the boiling/ steaming process. If you haven't oats to hand, you can also try doing the same with a fraction of the main grain you're cooking, effectively making a mix of whole grains and their own flour, which should yield the sticky properties you seek. As observed above, it won't turn out the same as rice (but then, nothing's quite the same as anything else), and you can experiment with different whole/ ground ratios till you find a balance you like.

Adding your freshly ground flour too soon will increase the risk of something burning on the bottom; too late, and it won't absorb enough water to cook properly - so the time at which you add the flour also affects the result.

I think I've also thrown a thinly-sliced potato into my rice cooker at some point, which made things hang together creamily. As does simply adding more water and cooking your grains for longer.

P.S. Are you sure the rice-in-arsenic thing is really a thing? I recall reading about this a while back, and I think the conclusion was pretty much the same as that regarding radioactivity in bananas - yes, it's there, but to a degree so infinitesimal in the context of all the other dietary and environmental sources of the thing we seek to avoid that it won't have a measurable effect. Unlike, for example, mercury in salmon. Every case is different, interesting, and merits investigation.