Would this return a party-member back to life from 3 failed death-saves?

Yes if it is within one minute of their last failed death save.  It doesn't do anything if they are just unconscious.  (IE between 0 hit points and less than 3 death save failures... If they are only unconscious they can just be healed with any heal spell, healing portion or stabilize with a medicine check.)

Thanks man, appriciate it! :)

What would happen to a vampire that I used this spell on?

Nothing, since they have been dead for longer than a minute :)

Quote from Rogue_Prince >>

What would happen to a vampire that I used this spell on?

If you follow Sage Advice, a just-destroyed vampire that has this spell cast on it will be revived as a vampire.

This is one of many, many reasons I no longer follow Sage Advice.

(If the vampire has not been destroyed, the spell will generally not do anything, since it can only target a creature that died within the last minute. If a creature were killed and then somehow instantly brought back as a vampire... maybe you could target it with revivify and turn it from a vampire into a living person at 1 hp? That'd be up to your DM. I'd probably say yes, just because it's a clever use of the spell and the circumstances are not likely to come up often.)

Do you DM's out there care about the 300gp Diamond component to use this spell?

I DM my group and for the majority of spells, i don't care if the players have incense or wires, or whatever material to channel their spells. But Revival spells (revivify, raise dead) seem more serious and have more of a need to follow the components necessary to do the spell.

Would it be a "bad-move" to incorporate that into my campaign? Thoughts?

No I think it's fine, it's understandable. I tend to do this for anything with a stated gp worth. Identify being one. You can make the choice to have it consumable and alter it how you want. You aren't limiting the player just following the rules 

I'm playing a cleric at a Tomb of Annihilation campaing, and I'm curious: how does this spell interact with the Soul Monger's curse? Will the other characters I revivify suffer from it and die again anyway?

From the Tomb of Annihilation introduction: 

• Any spell that breathes life into the dead (including revivify, raise dead, resurrection, and true resurrection) automatically fails if cast on a humanoid whose soul is either trapped in the Soulmonger or has been devoured by the atropal (see “Soul Devouring” below).
  
  

So, no, Revivify will not work in ToA.

GNOME

That was what I thought... Thank you!

Would gentle repose followed  by revivify work? (After the 1 minute)

Quote from Blobbytheepic >>

Would gentle repose followed  by revivify work? (After the 1 minute)

 

 Yes, according to Jeremy Crawford. It states specifically in the spell description that it can be used to extend the time limit on resurrection spells. 

"The spell also effectively extends the time limit on raising the target from the dead, since days spent under the influence of this spell don't count against the time limit of spells such as raise dead."

Strange question perhaps - What might your opinion be on using the spell to raise someone you had just killed through torture? Or basically raising someone that does not wish to be returned to life?  Since this isn't your standard type of resurrection spell, but almost more of a soul-defibrillator effect. 

Yes. If you're still making death saving throws then you're not dead and you would not use this spell. This is for people who have already died (failed all death saving throws and expired.) It must be used within 1 minute (10 rounds) of the last failed saving throw.

If I put the missing body part next to the body, will the spell be able to heal it?

The material components listed usually don't matter because nobody really takes a spell component pouch. You either need the component listed OR a spell focus UNLESS there is a gold cost to the component in which case the spell consumes the component and you can't use a spell focus.

For what it's worth (I've only just started playing DND recently so I might have missed something?) spells only consume the material component if it specifically says it does. This spell says it requires "diamonds worth 300 gp, which the spell consumes", while the spell Chromatic Orb requires "a diamond worth at least 50 gp". Chromatic Orb requires that you keep a diamond on you (which you would hold on to or whatever when you cast the spell), and can be reused every cast, while Revivify requires you to have a higher value of diamond(s) which are then consumed with every cast.

If I put the missing body part next to the body, will the spell be able to heal it?

No, Regenerate would be required for that.

If the player is infected with a necrotic disease when they die, would they still retain that upon reviving?