My little Piper does the same thing she will run and then fall asleep right after, shes very clean and somehow controls herself quite well, she never poops on the wheel she jumps out and goes right in the litter box then gets back on lol makes momma happy.. Oh her other favorite thing to do is when shes eating she decides shes sleepy and will just sleep in her little food bowl its adorable Ill get a picture up here soon

As for a mesh/wire wheel I wouldnt risk it, they can cause serious injury plastic wheels are much safer