Spinning Axe

Yes. The Spinning Axe will have two instances of the Temporary keyword - one that is active and one that is inactive. The inactive keyword will not trigger, but the active one will. Alas, Spinning Axe will die.

Multiple instances of the Temporary keyword are redundant per rule 816.2., but that rule refers to the fact that multiple instances that could trigger will only trigger once, as clarified by rule 816.2.a. In the case of granted Temporary on an attached Spinning Axe, that rule isn’t of any concern - there is only one instance of Temporary that could trigger, so it will.

It can help to visualize the attached permanent as it is undergoing these changes, as if it were being digitally simulated. Imagine all of its printed rules text as going gray while it is attached, as it is inactive. Then the granted Temporary keyword, because it is active, is the normal text color for Spinning Axe. There are two colors of Temporary on Spinning Axe, but the grey Temporary doesn’t have any function. We can treat it as if it didn’t exist for the purposes of rule 816.2.a.

You mentioned rule 187.4.c. earlier and it says “open state”, I don’t remember that in the old rules. What gives?

Control of battlefields cannot be lost while there are items on the chain, per rule 187.4.c. in the Unleashed rules update. This means that spells and abilities that banish and play units, or that kill units as a cost or effect in order to play another unit, will be able to kill or banish units at battlefields and then play the resulting unit to that same battlefield.

Let’s look at some examples:

I play Cruel Patron, and I have a 1 [M] Recruit alone at a battlefield I control. I choose to kill my Recruit to pay the additional cost of Cruel Patron. Although this causes a cleanup, because there is an item on the chain (Cruel Patron) I don’t lose control of the battlefield. I can choose to play Cruel Patron to that same battlefield with no issue.

I activate Baited Hook targeting a 4 [M] unit that is alone at a battlefield I control. When Baited Hook resolves, I kill that unit and look at the top 5 cards of my Main Deck. I choose to play a unit from among them, which goes pending on the chain. When Baited Hook’s activated ability resolves, the outstanding cleanup initiates, but I can’t lose control of the battlefield because the played unit is on the chain pending. Once the cleanup finishes, I will finalize the unit and I can choose the battlefield where I killed my 4 [M] unit to play it to.

I play Arcane Shift targeting my Thousand-Tailed Watcher who is alone at a battlefield I control. As it resolves, Watcher is banished and played to the chain. When Arcane Shift has finished resolving, a cleanup is outstanding and because Watcher is still pending on the chain, control of the battlefield can’t be lost. When finalizing Watcher, the battlefield I banished it from is a valid location to play it to because I still control it.

There is one final example worth discussing, though the interaction worked similarly prior to this rules update: in the previous rules update, if my Glasc Mixologist died alone in Combat at a battlefield I control, the active Combat at the battlefield prevented me from losing control of the battlefield. Thus I could play the unit from Mixologist’s Deathknell to the same Battlefield it died at. Previously this was only possible in Combat, but as of this rules update the same process will occur whenever Glasc Mixologist dies alone at a battlefield with the same controller, regardless of if there is a Combat there or not.