If you think about dying in terms of rounds and what that means for time, a death saving roll happens every 6 seconds. This means that a character knocked out has a minimum of 12 seconds (one roll of 1 for the double fail, and then a roll below 10) and a maximum of 30 seconds (2 successful rolls and 3 failed rolls) before they die (assuming no additional attacks and such).
Because this happens pretty quickly, I would keep turn order going after the combat is resolved. Your party might only have 6 or so seconds to get to each player and stabilize them. This can pose a challenge for the long ranged magic user to make it to the front to stabilize the fallen fighter, or vice versa. It's always possible they fail the stabilization check too, making for those 6-30 seconds to be pretty intense.
I find a bonus is that this additional last chance to die prevents some players from using kamikaze tactics where they think "who cares if I get knocked out? I'll still be stabilized right away." Not necessarily in my games.
In general...
The difficulty class ("DC") of the saving throw is based on the caster: 8 + proficiency bonus + casting ability modifier. (PHB p.205: "Saving Throws")
The GM rolls a d20 on behalf of the monster, adds the appropriate saving modifier based on the monster's stats, and compares to the spellcaster's save DC.
Your specific questions:
What are "any special modifiers?"
Things other than your ability score or proficiency that explicitly modify your saving throw. For instance, a Cloak of Protection will grant you +1 on all saving throws. (DMG p.159)
Is this example correct?
Yup.
To complete the example, if the monster was a Revenant, CR 5 would mean +3 proficiency so would add 3 to the d20 roll.
This is true if and only if the Revenant is proficient in that specific save. If you look at the stat block for a Revenant (MM p.259) you'll see saving throw modifiers listed for STR, CON, WIS, and CHA. You'll also see that the modifiers there are all three higher than the associated ability modifier. That's the proficiency/CR bonus, worked right into the stat block. (DEX and INT would save just as those ability modifiers.)
A worked example:
Your 6th-level, 16-WIS cleric Clare-Bear casts Sacred Flame at the zombie ZomTom. The description of Sacred Flame allows for a Dexterity save. (PHB p.272)
At 6th-level Clare-Bear's proficiency bonus is +3. With 16 WIS her spellcasting ability modifier is +3. Thus her save DC = 8 + 3 + 3 = 14.
ZomTom's DEX is 6, for a -2 modifier. He is not proficient in Dexterity saves(1). So ZomTom gets a net -2 modifier to its saving throw. (PHB p.311.)
So if the GM rolls a 16 or higher, achieving a 14 or higher after ZomTom's -2 DEX mod, he will have succeeded in his saving throw. (Ties always go to the roller of the d20; see PHB p.7, "The d20".)
(1) - Note that any proficiency bonus that the creature would gain from its challenge rating is already factored into the stat block. See, for instance, the Young Bronze Dragon on MM 108, whose CHA mod and CHA saving mod do not match. The dragon's proficiency bonus is added to the CHA mod as per "Proficiency Bonus by Challenge Rating" found on MM p.8. If there is no mention of an ability under "Saving Throws" then the creature is not proficient in that save.
Best Answer
A creature is absorbed by the Gibbering Mouth only if it's killed by its Bite attack, not when it drops to 0 hit points.
The Gibbering Mouther's bite attack states:
This means that the creature is not absorbed into the Mouther when it drops to 0 hit points. If the creature is dropped to 0 hit points by this attack, then it follows the usual rules (falls prone, is unconscious, and starts making death saving throws). If the Mouther attacks a creature at 0 hit points and makes it fail its 3rd death saving throw, or deals massive damage to the creature, killing it, only then it is absorbed.
Note: this is only usually relevant to player characters. NPCs and monsters typically die outright when dropped to 0 hit points (and thus would be absorbed when dropping to 0 hit points), unless the DM decides to have them fall unconscious and make death saving throws as well.