1. Can I combine Polearm Master/Sentinel with War Caster to cast spells as OAs when an enemy >enters my reach (PM) or when they Disengage from reach (Sentinel)?

(note: technically speaking, all of this will be up to your DM)

War Caster with Sentinel, creatures moving away even with Disengage, yes; creatures within 5' attacking a different target no: Sentinel specifies melee weapon attack.

War Caster with Polearm Master, RAI: no, see this link: Mearlsian Q&As

Do Polearm Master and War Caster combine to allow a magic user to make a spell >opportunity attack when they enter reach? No - polearm master applies only if you use the >weapons it lists to make the attack

2. If I want to maximise how often I can use my reaction to OA with spells, which is the >better pick, Polearm Master or Sentinel? On the face of it, it seems like the former, but >I've only started playing, so, not much experience yet.

If played/ruled RAI, then clearly Sentinel, but if Polearm Master/War Caster spell casting is allowed, it would still very much depend on the DM and campaign/adventures; for example many creatures with the ability to Disengage as a bonus action would heavily tip this in favor of Sentinel.

3. If Polearm Master is superior (as described above), is there a way to compare the >value of Sentinel's "stopping effect" (on a hit, you set the enemy's speed to 0) vs. the >added damage? The added element of battlefield control can be quite valuable (though hard >to quantify).

Again, this would be far too dependent on the DM/adventure to realistically quantify with any reasonable confidence.

RAW, it appears that using Booming Blade with Polearm Master and War Caster would work, but only with a Quarterstaff.

• Polearm Master causes an enemy to provoke an Opportunity Attack; not a "melee weapon attack" which would preclude spells.

• War Caster applies to Opportunity Attacks triggered by enemy movement; not "an enemy moving out of your reach", which would preclude Polearm Master's effect.

• Booming Blade has a range of 5ft, targets a single enemy, and uses the weapon that allowed Polearm Master to trigger. This satisfies all requirements of both feats, as well as Mearls' answer on Twitter.

• Finally, this only works with a Quarterstaff because other polearms viable with Polearm Master have a range of 10ft. When a creature enters that range and triggers the feat's Opportunity Attack, they would still be out of range of Booming Blade.

This appears to be implicitly confirmed by this Twitter exchange with Mike Mearls about whether Spell Sniper extends Booming Blade (it does, according to Mearls).