If the Cleric casts Mass Healing Word, targeting each of the 4 party members, how many times would the Cleric gain HP equal to 2 + Spell Level?
I would argue only once, because the trigger for Blessed Healer is "When you cast a spell...". You only cast one spell, so you'd only get healed once.
Taken further, would the Blessed Healer effect be prevented if the Cleric targeted himself with Mass Healing Word, in addition to the other 4 members of his party?
That's easier to answer. I'd say no, the effect is not prevented, since casting mass healing word to heal at least one person other than yourself clearly satisfies the requirement for Blessed Healer.
The Order Domain cleric's Voice of Authority feature (Guildmasters' Guide to Ravnica, p. 26) specifically says:
If you cast a spell with a spell slot of 1st level or higher and target an ally with the spell,
It doesn’t specify that the spell must have a positive effect, so any spell that targets a “creature” could be used to trigger the feature. It also doesn’t specify that the spell must target only an ally. On the subject of area of effect spells, the feature also says:
If the spell targets more than one ally, you choose the ally who can make the attack.
This means that spells that target multiple creatures are fine. So if a spell has multiple targets, the spell would trigger if one or more of them are allies, and if there are more than one, you choose which ally.
In the case of spiritual weapon, however, there's some ambiguity. According to Jeremy Crawford, lots of things can count as the "target" of a spell; this reddit post includes a good summary of his comments from this episode of the official D&D podcast, but in short, it's pretty broad. A spell like spiritual weapon initially targets a point, and creates the weapon there, but then the weapon - and by extension the spell - targets creatures for attacks.
That sort of usage is subject to interpretation by your Dungeon Master, who will probably be guided by the fairly clear intent of the spell - that it's an extra bonus for allies buffed by a spell cast by the cleric, rather than usable with a spell intended to attack enemies for many rounds, allowing the Order cleric to grant multiple additional attacks at the cost of a single spell slot. However, it is probably allowed by the rules as written, especially when Jeremy Crawford's comments are taken into account.
I suspect the language is a little loose to make sure it allows for all buffing spells, some of which might also have a delay between casting and targeting an ally - hence “if you cast a spell” rather than the perhaps clearer “when you cast a spell”.
Best Answer
The Order Domain cleric's Voice of Authority feature (GGtR, p. 26; TCoE, p. 31) says:
If your ally, the intended target of healing word, is at 0 HP and making death saving throws, they're unconscious and presumably prone. Voice of Authority triggers after you cast healing word on them. When you cast healing word on them, they regain an amount of HP equal to 1d4 + your spellcasting ability modifier, which causes them to no longer be unconscious.
Since they are no longer unconscious, they can take actions and reactions as normal (i.e. actions on their turn, or reactions when their trigger occurs), assuming they're not affected by another condition or effect that prevents them from doing so. Thus, the ally you healed could indeed use their reaction to make a weapon attack against a creature of your choice that you can see, as long as the creature you choose is within range of the ally you healed.
As Miniman notes in a comment on Derek Stucki's answer, though, the attack would be made at disadvantage due to the character still being prone. (The character would continue being prone until at least their next turn; on their turn, they can use an amount of movement equal to half their speed to stand up.)
In addition, when the character was reduced to 0 HP, they would have fallen unconscious (which is also what causes them to fall prone). As a result, they would have dropped whatever they were holding, including a drawn weapon. The character would not yet be able to pick any such weapon back up (or draw a new one) after being healed – since they can only do so on their turn, using a free object interaction or the Use an Object action. Thus, the only attack they can make as part of this process is an unarmed strike or an attack with a natural weapon.