[RPG] What are a magic item’s saves

Proving a negative is impossible, of course, but I'm fairly confident that there are no general guidelines for creating magical items with triggers other than the main four from the question:

• Spell completion
• Spell trigger
• Command word
• Use activated

... plus continuous items.

However, the Contingency spell would probably be useful for the "when X happens, do Y" items.

For pricing such items, the only real rule is to find an existing item of similar power and price your item similarly. The Item Creation Rules have guidelines, and there's some evidence that an item that requires the "right" slot to duplicate a feat costs around 5k (eg., the Gloves of Arrow Snatching, which allow use of the Snatch Arrow feat twice per day), but the rule is to price it based on similarly-powerful items.

I would specifically recommend asking whether a generic character would buy the new item instead of a +X weapon, armor/shield, Stat item, or cloak of resistance: if a generic character is definitely going to buy the item instead of a +2 shield, the price should be higher than the 4k a +2 shield would cost; if it's a toss-up between the new item and a +1 weapon, a price of around 2k is probably about right. This is difficult, especially for new GMs or items that are highly situational (eg., the "self-destruct if a person is too far away from an item" item) or setting-dependent (eg., the auto-healer if divine magic is rare in "this" world).

In my experience, it's generally been best to err slightly on the side of making the item too expensive, then adding some extra treasure if need be.

Remember, too, that "Activating a magic item is a standard action unless the item description indicates otherwise." (Using Magic Items) The rules don't preclude magic items that can be activated as move/swift/free actions, they just don't provide much in the way of guidelines for making them.

That said, it's entirely possible to build most of the items in the question's list using just the Big 5 triggers:

• Self-destruct device if a person forgets some important item somewhere
  • continuous item using Alarm and/or Scrying as a base requirement, plus a sufficiently powerful boom-ey spell
• Magical flashlight
  • use activated or command word item based on several of the Evocation (light) spells, eg., light
• Booby-trapped bags of holding
  • again, adding Alarm to the base requirements could serve to trigger the booby trap, as could simply calling it a variant Bag of Devouring
• Device that casts True Sight on owner when near evil-aligned characters, or perhaps Hide Alignment near good-aligned characters
  • again, a continuous item with Alarm and the appropriate Detect spell (eg., Detect Evil)
• A crossbow that automatically fires a bolt when [given target] is in its cross hairs
  • this is a reasonable interpretation for how True Strike works; an item that allows the wielder to use True Strike with a single attack from a weapon could be said to do this; similarly, Haste grants an extra attack in some situations...
• Necklace that casts a buff when its owner falls prone
  • a constant item with Contingency as part of the creation process could probably do this
• Automatic potion-imbibing devices when its owner is low on HP.
  • again, Contingency could probably be used in creating this

The easiest way to gain advantage on a death saving throw is to use inspiration.

If you have inspiration, you can expend it when you make an attack roll, saving throw, or ability check. Spending your inspiration gives you advantage on that roll.

Alternatively, you could be a Wild Magic Sorcerer, and use Tides of Chaos. Well, probably.

Starting at 1st level, you can manipulate the forces of chance and chaos to gain advantage on one attack roll, ability check, or saving throw.

The 3rd- level spell Beacon of Hope explicitly grants advantage on death saving throws for its duration.

Fighters can use Indomitable to reroll a failed saving throw - not precisely the same as advantage, but pretty close.

Beginning at 9th level, you can reroll a saving throw that you fail.

There are almost certainly more options that I've missed - getting advantage on things is a pretty fundamental part of 5e.