[RPG] Interaction of Invoke Duplicity and Misty Step

Your main question: Can Invoke Duplicity use touch spells? YES

And: Could a cleric use this as a way to heal allies with Cure Wounds up to 120 feet away? Or use Inflict Wounds on enemies in the same way? YES

Invoke Duplicity (PHB p.62)

For the duration, you can cast spells as though you were in the illusion's space, but you must use your own senses.

This means you literally can cast spells as if you were standing where the illusionary duplicate is, with the exception that you must be able to target the spell using your own senses.

In the case of the Cure or Inflict Wounds spells, for instance, if you can target the creature you want to heal or hurt using your own senses (you can see them, or can touch them yourself if you are blinded somehow) then yes you can cast the spell.

The Cleric, needs to be in range for the spell to work, but you can cast counterspell if you see either of them casting a spell:

Counterspell has a casting time of:

1 reaction, which you take when you see a creature within 60 feet of you Casting a Spell

So you would think that you would need to be able to see a real creature casting a spell to have this work. However, Jeremy Crawford, 5e lead rules designer, responded on twitter to two questions about this topic on twitter.

Q1: Hi!! Quick q: Trickery cleric uses Invoke Duplicity to cast spells from. The illusory image that ID [sic] creates is then counterspelled in the process of casting a spell; does the counterspell stop the spell cast, or fail as it's an illusion?

A1: Invoke Duplicity doesn't protect a cleric from being counterspelled if the person casting counterspell can see the cleric/illusion and the cleric is within 60 feet of the caster.

source

and

Q2: Does seeing the illusion from Invoke Duplicity count as “seeing” the actual Cleric?

A2: Seeing the illusion created by Invoke Duplicity isn't the same as seeing >the cleric. However, seeing the illusion is sufficient for some things in the >game, like counterspell, that are triggered by seeing processes (seeing the >cleric casting a spell, for example).

source

Based on this ruling, if you can see either the illusion or the cleric themself in the process of casting a spell, you have a valid trigger for counterspell. This is even though the illusion isn't a creature. With that being said, Invoke Duplicity allows you to cast the spell as if you were in the illusion's space, but the Cleric is still the one casting the spell. Even if you react to the illusion appearing like it's casting a spell, the cleric needs to be in that 60 foot range for the counterspell to be successful, since they are the one who is going to be counterspelled.