[RPG] What rules govern group skill checks

Rules for group checks in DnD 5e can be found on page 59 of the player's basic rules or page 175 of the PHB and are as follows:

To make a group ability check, everyone in the group makes the ability check. If at least half the group succeeds, the whole group succeeds. Otherwise, the group fails.

Do all opposing skill checks automatically succeed on a natural 20 ( unless there's a tie )?

No, but it's pretty hard for an opponent to beat a natural 20. The opposing roll only ever wins if its total check is higher despite the natural 20. This may happen, say, when a player has a negative modifier to a roll and the opponent rolls high and/or has high scores in the opposing roll.

In one of our games, my 8 strength Lore Bard beat our 18 strength fighter in an arm-wrestling, basically a Strength(Athletics) contest, by picking up Expertise in Athletics, throwing insults at him (Cutting Words Lore Bard feature), and cheating (casting Enhance Ability on myself).

If not all opposing skill checks automatically succeed on a natural 20, do any of them? If so, which ones?

Ability Checks don't have critical success unless house-ruled into your game but I'd be wary of this and that's another topic entirely.

Besides saves & attack rolls, what treats a natural 20 as an automatic success?

Only Attack Rolls treats a natural 20 as a critical hit, even if you roll a natural 20 on a Wisdom Saving Throw, you're still going to be frightened of that dragon if the total doesn't equal or surpass the Save DC.