When Mobius is summoned, and he gets ringed(directly after summoning), does his magic/trap removal effect work or no?

Yes, but if RoD was one of them I guess it's more like 1 would be destroyed.
His effect would be link1 on the chain and Rod would be link 2.

"Mobius the Frost Monarch" has a triggered effect. When he is summmoned, the effect automatically goes off and you choose up to 2 S/Ts on the field to destroy. THen you pass priority to your opponent to respond to the summon. Your opponent will then be able to chain to the effect, even if the chain involves the cards targetted by "Mobius the Frost Monarch", in this case, "Ring of Destruction" was one of them.

The chain would resolve backwards

Mobius --> Ring

Ring would destroy Mobius and the the effect of mobius would destroy the 2 S/Ts targetted.

Mobius' effect has an automatic Priority. You may respond to it in any way you see fit.

Unless the card specifically states it stop effects,(like Skill drain or Divine Wrath) no matter what you activate will not stop Mobius from carrying out it's effect.

Step by step:

Summon Mobius
Player Must choose 2 Mt to destroy now
Chance to chain here
Resolution of chain, the 2 mt are destroyed now(or any ones that werent activated out of the targeted 2)

The only way a monster is NOT successfully Summoned, Flip or Special Summoned is if the summoning itself is negated like with Horn of Heaven, Solemn Judgement of Royal Oppression (for special summon) since, if the summon is negated, there, technically, was no summon to begin with so the monster was never summoned.

There are only three cards in YGO that can negate a Monarch's effect.

1. Solemn Judgement. This negates a m/t/summon. Thus Monarch won't go off.
2. Horn of Heaven. Negate a Summon.
3. Divine Wrath. Negates an effect monster.

All other cards like "Trap Hole", "Bottomless Trap Hole", "Torrential Tribute", and "Ring of Destruction" will NOT negate Monarch effects as they destroy the monster after it summons successfully.

I get into that fight alot... sad to say. As people think Trap Holes negate summons.

Missing one. Skill drain, when chained AND when on the field before hand negates the effect quite nicely actually.

The only thing is it's the least aggressive of them.

Ok I understand, the only part that confused me was the Tribute Summoned Sucessfully... is successfully just getting on the field without Solemn Judgment/Divine Wrath/Horn of Heaven working on it?

Right, new I was missing one, and it wasn't Opression since Mobius is effectless on a Special, thanks. The thing is, whenever someone says "Mobius was destroyed, he gets no effect" if you tell them the rules and they keep arguin', they probably know the truth and are trying to pull one on you.

Monsters DO NOT have priority, effect or not.

You as the turn player have priority

Mobius' effect won't resolve in two cases:

His summon was negated: Solemn Judgement, Horn of Heaven.
His effect is negated: Divine Wrath, Skill Drain.

In ALL other cases of chain, his effect resolves even if he is destroyed, his effect is treated as a Trigger effect, triggered by his Successful Tribute Summon.

There is one other case according to the UDE site.