Does The Ur-Dragon make it self cheaper when I play it? I mean, it’s not in play or in the commander zone when it’s on the stack right? I’m just a bit confused about this.
No. As you pointed out, it is on the Stack so it can't be on the battlefield or in the Command Zone. However, even easier is that Ur-Dragon explicitly says "other Dragon spells" (it would still work the same without this but it makes it clear he can't apply to himself).
Does The Ur-Dragon make itself cheaper?