It is late at night and I am running on empty but I could really use an answer by morning
Do the Oath Against Savagery and Sacred Servant archetypes stack?
They probably stack. There is the possible interpretation that sacred servant adding domain spell slots counts as modifying the "spells" class feature, which would make these not compatible.
Sacred servant adds domain spell slots, oathbound paladin adds oath spell slots which serve the same purpose. I think you could technically have both unless they're both thought to alter 'spells' (which would rule out the combo), but IMO treating the domain/oath spell slot as the same thing and requiring the paladin to prepare only one extra spell slot (choose domain or oath spell) per spell level is the right way to do it.
Oathbound paladins add spells to the class list, they do not add a bonus slot like domains do.
Oathbound paladins add spells to the class list, they do not add a bonus slot like domains do.
This is where is gets confusing because it makes it clear that Oaths stack but then confuses things with the domain reference
A paladin’s oath influences what magic she can perform. An oathbound paladin adds one spell to the paladin spell list at each paladin spell level she can cast (including spell levels for which she would only gain spells per day if her Charisma were high enough to grant bonus spells of that level). Her oath determines what spell is added to the spell list. If the paladin has multiple oaths, the spells from each oath are added to her spell list.
If an oathbound paladin has more than one oath, she may prepare any one of her oath’s spells in that slot (similar to a cleric choosing one of her two domain spells to prepare in a domain spell slot).
You're right, no slot, I misread that. Since they're affecting quite different things (the spell list for the main slots vs. domain slots only) they probably should stack.