You can activate the effect of
Snipe Hunter and target your opponent's Ring of Destruction. If they want to use it, they have to chain it now. Either way, when
Snipe Hunter's effect resolves, you do the die roll and destroy the target card (if the roll was successful).
The point is, it's not certain whether the target card will be destroyed or not by the time you get to chain cards. Usually you'd chain the card that's about to be destroyed in order to get its effect, even if it's not the best time, because otherwise you'd just lose it.
Of course, when
Snipe Hunter's effect finishes resolving, your Ring of Destruction is in your Graveyard, so you can't activate it. But there's always a chance (1 in 3) that
Snipe Hunter's effect won't destroy the card, in which case you can activate it afterwards (if you haven't already).
Solemn Judgment can negate Spell Cards, Trap Cards, Normal Summons, Flip Summons, and built-in Special Summons (i.e. "this is how you Summon me" Special Summons - think Cyber Dragon/The Fiend Megacyber/Chaos Sorcerer/etc.).
Since it can only negate built-in Special Summons, it cannot negate Special Summons caused by other card effects. It can, however, negate that other card (as long as it's a Spell or Trap Card).
So in the case of
Advanced Ritual Art, you need to negate the Spell Card, because you cannot negate the Special Summon alone (because you can't activate cards during the resolution of a chain, which is when the Special Summon takes place - during the resolution of
Advanced Ritual Art).
The Tribute Summon of Mobius the Frost Monarch WAS negated by Solemn Judgment, because it was a Tribute Summon. I assume you were talking about its Tribute Summon here.
Someone else will probably come along and explain it much better than I have (and with more emphasis on not being able to activate Solemn Judgment during a card resolution).
Conjecture: Would I be right in thinking Solemn Judgment can negate Spells and Traps, and any Summon that is not a part of a chain? It makes a much easier explanation if so.