Before Beaver adopted his monkey and rented a parrot Wally wanted both. In Beaver Takes a Bath Wally asks Ward if he can buy a monkey. In Baby Picture he tells June he might get a parrot for his cage.

He also asks June and Ward if he can grow a mustache but later he just does it. He never wants a burro. He asks June if he can take a cross country trip this summer and Beaver got to do that in season 6.

Sounds like later writers took Wally's dialog and ran with it, for some Beaver stories.

Wally wanted to go to Mexico but they never sent Beaver there.

That’s right and Beaver never ran away to Mexico either. Wally wanted to get tattooed but Beaver never wanted a “Mother” tattoo. I think Wally asks if he can quit school in one episode and get his diploma through the mail but the only one who did that was Eddie.