When a PC finds a +1 weapon, what stat or stats increase by one? For example, in the Lost Mine of Phandelver adventure in the Starter Set, there is a +1 Longsword. What does it do?
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.
Yes, sort of, but damage works a little differently in this edition of D&D
Magical describes the weapon, and the attack
The magical sword makes a magical attack, it doesn't do magical damage per se since "magical" isn't a damage type of its own. This is a subtle point, but it is worth remembering. (The related case of silvered weapons follows this same rule / logic).
... a magical attack is an attack delivered by a spell, a magic item, or another magical source (Basic Rules, DM, p. 4; under Vulnerabilities, Resistances, and Immunities).
You describe Talon as a longsword, +1. In the DMG, you will find such weapons under Magic Items, in the category of Weapon, +11.
In regards to a creature that has "resistance to damage from non-magical weapons", does Talon deal magical or non-magical damage?
You cited the standard rules block incorrectly.
Damage Resistances bludgeoning, piercing, and slashing from nonmagical attacks (Magmin; MM)
Damage Immunities bludgeoning, piercing, and slashing from nonmagical attacks not made with silvered weapons (Werewolf; MM)
Unlike some video games and some other RPGs, D&D 5th edition does not specify magical as a type of damage, although Force as a damage type is described as pure magic that does damage.
Damage Types (Basic Rules, p. 75)
Damage types have no rules of their own, but other rules, such as damage rresistance, rely on the types. The damage types follow, with examples to help a DM assign a damage type to a new effect.
The rules cite the following kinds of damage. (edited for brevity)
Acid. Bludgeoning. Cold. Fire. Force. Lightning. Necrotic. Piercing. Poison. Psychic. Radiant. Slashing. Thunder.
Note that 'magical' isn't one of the listed damage types. Long swords do slashing damage. (Weapons table, PHB).
Weapons typically do Bludgeoning, Slashing, or Piercing damage. Monster stat blocks often show resistance or immunity to Bludgeoning, Slashing, or Piercing damage from non-magical, or non-silvered, weapons. (Werewolf, magmin, as above). Some also show resistance or immunity to other damage types. A red dragon, for example, is immune to fire damage. (MM; SRD p. 286 also) Fire is a damage type.
How does this work during play?
Ask yourself: is the creature resistant to, or immune from, non-magical attacks?
If yes, then a non-magical weapon does half damage (resistance) or no damage (immunity), but a magical weapon overcomes that resistance or immunity and therefore still does full Bludgeoning (mace), Slashing(sword), or Piercing(spear) damage to the target; it's attack is magical.
What damage does Talon do?
Talon is a magical sword (+1); it makes magical attacks and thus bypasses resistance or immunity to weapon based slashing attacks, doing full slashing damage.
Put another way, the sword isn't doing "magical damage" (type) but is rather doing a type of damage because it is making a magical attack.
Caveat: against Swarms, the slashing coming from a magical attack, or not, won't matter. Swarms have:
Damage Resistance: bludgeoning, piercing, slashing (MM)
Note that it doesn't specify a source, the way a werewolf does.
Damage Immunities: bludgeoning, piercing, and slashing from nonmagical attacks not made with silvered weapons (MM)
1 Under "Magic Items" we find.
Weapon, +1, +2, or +3
Weapon (any), uncommon (+1), rare (+2), or very rare (+3)
You have a bonus to attack and damage rolls made with this magic weapon. The bonus is determined by the weapon’s rarity. (DMG, p.213)
"Magical Attacks" as a game feature.
A good example of this is the Monk's 6th level ability Ki-Empowered Strikes
Starting at 6th level, your unarmed strikes count as magical for the purpose of overcoming resistance and immunity to nonmagical attacks and damage.
The attack is magical, which then applies damage.