
Claxon |

Not officially.
UMD can let you emulate an ability score, a race, an alignment, or class ability. Worship of a deity isn't an option to emulate.
Now could you argue that it's reasonable? Maybe. What would the DC be? I'm not sure. I would probably set in higher than any of the published DCs, so like 35.
Of course, looking at that particular item I would tell you no if I was your GM. I don't like anything that breaks WBL guidelines, which is what that item does.

Azothath |
no
see B.Obj & UMD thread, 2023
so again, not on the list AND emulating an alignment is a 2nd check.
again, a Home GM could allow this as it's not unreasonable but RAW does not set a DC.
Many of these restrictions on Magic Items are modeled as discounts (see creating Magic Items big table and various class and alignment restrictions detailed afterwards). Their DC should follow that discount as that makes good design sense.
Droskar's Guiding Ring[ring] $2600
IMO these also provide a good follow-up plot consequence as divine entities tend to discover perversions and misuse of their granted powers. Don't worry - Droskar seems pretty forgiving... <EG> do it.. you know you want to... and again, yes, toil leads to success!

Mysterious Stranger |

For the most part this is not going to work. RAW a worshiping a deity is not a class feature.
If the setting for the campaign requires divine spell casters to serve a specific deity, it could be considered a class feature. This would be in the realm of a house rule so it would be up to the GM to decide. Personally, I would not have problems with it and do require most divine spell casters to serve a specific deity. Oracles and druids or druid variants are the exceptions in my campaigns.