What weapons are subject to ancestries’ weapon familiarity proficiency changes?

Consider Dwarven Weapon Familiarity.

A dwarf that takes it becomes trained in the battleaxe, pick and warhammer. Also from this feat:

You also gain access to all uncommon dwarf weapons. For the purpose of determining your proficiency, martial dwarf weapons are simple weapons and advanced dwarf weapons are martial weapons.

Does this part include the weapons that don’t have the Dwarf trait, but are mentioned above? Or does it only apply to the clan dagger and Dwarven war axe?

If it only applies to weapons with the respective ancestry’s trait, then it seems like certain ancestries have been stiffed by this. The hobgoblin equivalent of this feat, for example, mentions martial hobgoblin weapons being treated as simple, etc. But there aren’t any weapons with the hobgoblin trait, meaning the feat is currently useless (aside from giving the appropriate training in the listed weapons).

tl;dr are the weapons that don’t have an ancestry trait included in the weapons whose proficiency is treated as lower (advanced -> martial, martial -> simple) for a character with the feat?

For a basic unarmed strike, is only the proficiency bonus included in the attack roll, or is the Strength modifier also added?

The Player’s Handbook, Chapter 9, states about attack rolls:

The ability modifier used for a melee weapon attack is Strength

A bit later on it also states about melee attacks:

Instead of using a weapon to make a melee weapon attack, you can use an unarmed strike: a punch, kick, head-butt, or similar forceful blow (none of which count as weapons). On a hit, an unarmed strike deals bludgeoning damage equal to 1 + your Strength modifier. You are proficient with your unarmed strikes.

I’m not sure whether an unarmed strike is considered a “melee weapon attack.”

For a basic unarmed strike (i.e. no Monk, Tavern Brawler or anything else that enhances unarmed strikes), is only the proficiency bonus added to the attack roll, or is the Strength modifier also added?

Clearly the Strength modifier is added to the damage, but I’m asking about the attack roll.

What is the point of a shield proficiency?

PHB describes the downsides of wearing an armor you are not proficient with:

Armor Proficiency.
Your class gives you proficiency with certain types of armor. If you wear armor that you lack proficiency with, you have disadvantage on any ability check, saving throw, or attack roll that involves Strength or Dexterity, and you can’t cast spells.

However, a shield is not armor.

The Spellcasting chapter says about armor explicitly, and not shields:

Casting in Armor
Because of the mental focus and precise gestures required for spellcasting, you must be proficient with the armor you are wearing to cast a spell. You are otherwise too distracted and physically hampered by your armor for spellcasting.

It seems a Sorcerer can easily wield a shield. What exactly are the downsides of not being proficient with shields?

Does/Should the Gladiator background grant proficiency in the weapon it gives?

The Gladiator background (an Entertainer variant, from PHB p. 130-131) gives a strange weapon (like a trident or net) as starting equipment, but doesn’t explicitly specify that it gives proficiency with said weapon.

Is it implied that it does? (Or if not, should it, considering that a seasoned gladiator is expected to be particularly skilled with his weapon of choice?)

Does Natural Explorer’s double proficiency apply to perception checks while keeping watch?

The Natural Explorer class feature is sort of vague about what skill checks count

When you make an Intelligence or Wisdom check related to your favored terrain, your proficiency bonus is doubled if you are using a skill that you’re proficient in.

I emphasized related to, because it doesn’t say "while in," so I’m wondering whether keeping watch in your favored terrain counts as related to your favored terrain. This could represent something like being more familiar with what’s naturally around, and so you have an easier time spotting what shouldn’t be — i.e. the approaching enemy.

Does it apply?

Does a player need to perform a successful tool check with their Mason’s Tools proficiency to benefit from the Demolition feature?

The entry for Mason’s Tools within the PHB (p154) lists this as one of the proficiency’s benefits:

Demolition. Your knowledge of masonry allows you to spot weak points in brick walls. You deal double damage to such structures with your weapon attacks.

It then lists this as one of the possible activities:

Find a weak point in a stone wall [DC 15]

One might interpret this to mean that you must make a successful DC15 check with the Mason’s Tools before being able to deal double damage to a stone structure with your weapon attacks.

Another might read these as isolated entries, believing that your proficiency with Mason’s Tools automatically allows you deal double damage to stone structures with your weapon attacks as a passive benefit. Meanwhile, successfully performing the tool check above bestows its own separate benefit.

Obviously, the confusion lies in their using the phrase "weak point" in both the Activity as well as the Demolition feature. However, if read literally, the Demolition feature does explicitly state "You deal double damage to such structures…" and not "you deal double damage to weak points within such structures…".

So, to word this as questions:

Do you need to perform to a successful tool check to benefit from the Demolition feature of the Mason’s Tools?

And if not, what would be the benefit of finding a weak point in a stone wall with a successful check of the Mason’s Tools? (For example: Would a weak point within a stone structure simply have fewer hit points than the rest of that structure?)

Utility of Proficiency Bonus in Statblock for Summoned Creatures without Proficiencies?

Looking into Tasha’s Cauldron of Everything, I was reviewing the stat block of the Fey summoned by the Summon Fey spell. Amongst the statistics provided is the Proficiency Bonus of the spirit, defined as "equals [the caster’s] bonus."

The Summoned Fey has no Skill, Tool, or Saving Throw Proficiencies. Presumably the Proficiency Bonus value is already effectively baked into the Shortsword Attack, and the Spell Save DC for its bonus actions, as those are based on the Caster’s Spell Attack and Spell Save DC.

Is there another reason why the Player or DM would need to know this Proficiency Bonus, or is it merely a leftover from using a similar Stat Block for companions that do have proficiencies, such as the Battle Smith’s Steel Defender, or the Homunculus Servant Infusion?

What’s the use for proficiency with a gaming set?

Compared to other skill/tool proficiencies, being proficient with a gaming set seems horribly underpowered. Especially since there’s even a subdivision into different games.

I expect such games to come up sometimes, to win a McGuffin or gain trust. But the adventure needs to be tailored to it, unless you run some kind of shameful Yu-Gi-Oh ripoff 😛

Is this just a horrible Character building choice, or are there hidden used I’m overlooking?