I think the twin choosing an axe was to build suspense versus what you propose, which I guess is that the writers somehow painted themselves into a corner and needed a quick and neat resolution. Not likely considering it would have been much easier to re-write in Hank shooting one less bullet so he could defend himself against the twin. Deus ex machina is commonly seen in movies and series at their conclusion when a "god from a machine" is needed to save the writers from the corner they've written themselves into. In a single scene, such as the one we're discussing, if the writer needs to be saved from himself not a lot needs to be un-done to do it. It would take much more writing, in this case, to cop out then to just re-write a few things.
Regarding your other examples: I think you more or less just want to know the potential characters in play before it's ever really needed. That's fine and all but it's not deux ex machina when these characters are introduced, even if yes, they are used to resolve something. In the case of the fake Heisenberg, for example, I thought it was both a comedic resolution and an apt display of sleazy Saul's connections. Personally, I have no heartburn with it. I think this might be lost on you potentially because you might have recently learned what deus ex machina is and are eager to use it as often as possible.
You're in a universe where a chemistry teacher with a DEA agent brother-in-law with a hard-on for meth cooks chose to go into the meth making business. If you can believe this, you can believe there's a guy that goes to jail for a living. You can't really pick and choose what to bitch about and slap the deux ex machina label on everything. I mean, you can, it just doesn't mean it's right.
