Yes, the combination "quarterstaff + War Caster + Polearm Master" allows you to cast Booming Blade when someone enters your reach.

A comment by Mike Mearls from 2014 states that you are required to make the opportunity attack with an actual polearm - wielding a polearm in one hand doesn't allow you to make the polearm master opportunity attack with a non-polearm in your other hand.

No - polearm master applies only if you use the weapons it lists to make the attack.

In 2016, he came to the conclusion that combining Polearm Master, War Caster and Spell Sniper allows casting Booming Blade when someone gets within 10 ft. of you. This fits within the view that an attack with the polearm listed weapon would need to be part of the spell for it to work.
Note that Mearls rulings aren't considered official, only Jeremy Crawford's are, and RAW, you don't have to use a polearm for the attack.

Anyhow, without the feat Spell Sniper or another feature that increases Booming Blade's range to 10ft (such as a Sorcerer's Distant Spell metamagic option), you can only combine Booming Blade with a quarterstaff if you want the polearm master opportunity attack.
This is because the feat says "You can keep your enemies at bay with reach weapons". This implies that letting them get close to 5ft, even though your reach is 10ft, is against the wording of the spell since you're not keeping them at bay with a reach weapon in that case.

Using Spell Sniper allows you to extend the range of Booming Blade to 10 ft, which allows you to satisfy both requirements. You are keeping them at bay, and you are using a reach weapon listed in the feat.

Yes, the combination "quarterstaff/spear + War Caster + Polearm Master" allows you to cast Booming Blade when someone enters your reach.

A comment by Mike Mearls from 2014 states that you are required to make the opportunity attack with an actual polearm - wielding a polearm in one hand doesn't allow you to make the polearm master opportunity attack with a non-polearm in your other hand.

No - polearm master applies only if you use the weapons it lists to make the attack.

In 2016, he came to the conclusion that combining Polearm Master, War Caster and Spell Sniper allows casting Booming Blade when someone gets within 10 ft. of you. This fits within the view that an attack with the polearm listed weapon would need to be part of the spell for it to work.
Note that Mearls rulings aren't considered official, only Jeremy Crawford's are, and RAW, you don't have to use a polearm for the attack.

Anyhow, without the feat Spell Sniper or another feature that increases Booming Blade's range to 10ft (such as a Sorcerer's Distant Spell metamagic option), you can only combine Booming Blade with a quarterstaff or a spear if you want the polearm master opportunity attack.

This is because the feat says "You can keep your enemies at bay with reach weapons". This implies that letting them get close to 5ft, even though your reach is 10ft, is against the wording of the spell since you're not keeping them at bay with a reach weapon in that case.

Using Spell Sniper allows you to extend the range of Booming Blade to 10 ft, which allows you to satisfy both requirements. You are keeping them at bay, and you are using a reach weapon listed in the feat.

Yes, if the spell includes an attack from a weapon listed and at the correct range
Mearls' is the only comment from WotC I can find directly on this topic.

No - polearm master applies only if you use the weapons it lists to make the attack
-Sept 2014

Then in 2016 he came to the conclusion that the Polearm Master / War Caster / Spell Sniper / Booming Blade combo could work to cast booming blade at 10 ft.
  This fits within the view that an attack with the polearm listed weapon would need to be part of the spell for it to work.

Strictly RAW, there is no restriction for the polearm weapon to be used for the attack, but as you've pointed out, the only comment from a game designer says that is the intent.

Using Booming Blade as an example, the range is 5 ft. So an attack "when they enter your reach" cannot include Booming Blade since they enter your reach at 10ft and you cannot cast the spell there. And since the text from the feat states "You can keep your enemies at bay with reach weapons.", it seems that letting them close to 5ft is against the wording of the spell, that is specifically not keeping them at bay with a reach weapon.

Using Spell Sniper allows you to extend the range of that spell to 10 ft, which allows you to satisfy both requirements. You are keeping them at bay, you are using the reach weapon, you are using a weapon listed in the feat.

Yes, the combination "quarterstaff + War Caster + Polearm Master" allows you to cast Booming Blade when someone enters your reach.

A comment by Mike Mearls from 2014 states that you are required to make the opportunity attack with an actual polearm - wielding a polearm in one hand doesn't allow you to make the polearm master opportunity attack with a non-polearm in your other hand.

No - polearm master applies only if you use the weapons it lists to make the attack.

In 2016, he came to the conclusion that combining Polearm Master, War Caster and Spell Sniper allows casting Booming Blade when someone gets within 10 ft. of you. This fits within the view that an attack with the polearm listed weapon would need to be part of the spell for it to work.
Note that Mearls rulings aren't considered official, only Jeremy Crawford's are, and RAW, you don't have to use a polearm for the attack.

Anyhow, without the feat Spell Sniper or another feature that increases Booming Blade's range to 10ft (such as a Sorcerer's Distant Spell metamagic option), you can only combine Booming Blade with a quarterstaff if you want the polearm master opportunity attack.
This is because the feat says "You can keep your enemies at bay with reach weapons". This implies that letting them get close to 5ft, even though your reach is 10ft, is against the wording of the spell since you're not keeping them at bay with a reach weapon in that case.

Using Spell Sniper allows you to extend the range of Booming Blade to 10 ft, which allows you to satisfy both requirements. You are keeping them at bay, and you are using a reach weapon listed in the feat.