As Rhadamanthus explained, one damage event won't make this work. What you could use is something like Toralf, God of Fury  ^Flip or Boros Reckoner, both of which are triggered abilities, to create multiple damage instances so the damage will be multiplied each time. For example, if you have Furnace of Rath in play and target Boros Reckoner with Shock, the damage dealt to your Reckoner will be doubled from 2 to 4, then its ability will trigger and deal 4 damage to any target, which will be doubled again to 8.

You will get a copy of Kefka, Ruler of Ruin  ^Flip. Creating a token copy of a transforming double-faced permanent will create a transforming double-faced token with the current face up (see Rule 707.8a).

This doesn't work the way you hope, for two reasons.

To start with, yes, you can absolutely do this. You cast Mirror Image, which has the type Shapeshifter. This triggers Volo, Guide to Monsters if you don't have any Shapeshifters in play or in your graveyard, copying the spell. If no one responds, you'll respond the copied Mirror Image, which will create a token Mirror Image that can enter as a copy of any creature you control, then the original spell will resolve and Mirror Image will enter (again, can enter as a copy).

On the battlefield, the Mirror Images will be whatever they copied, and will therefore enter as something other than Shapeshifters unless you decided for some reason not to copy anything. This doesn't matter to Volo, because he triggers when the spell is cast, not when the spell resolves and the creature enters the battlefield.

With all that said, here are the two issues:

First, Volo triggers only on cast. Having a second Volo when your Mirror Image original (not the copy) enters won't do anything because it's not being cast, it's entering the battlefield. This would only work if you already had two Volos when you cast Mirror Image.

Second, however, Volo is a legendary creature. If at any point you control two legendary creatures with the same name, you must sacrifice all but one of them as a state-based action. At the moment Mirror Image enters as Volo, you will have two copies of Volo (and this could potentially trigger enters-the-battlefield abilities of something), but immediately after that, before anyone can respond or anything can resolve, you will have to pick one Volo and sacrifice the rest of them.

The only way around this kind of thing is a clone like Spark Double, which can copy legendary creatures without issue because it removes the legendary typing when it copies something, or another way around the legend rule like Mirror Gallery or Sakashima of a Thousand Faces.

As a side note, Judith's best friend is Electrickery. Who says Izzet and Rakdos can't get along?

Well, yes, but actually no, for two different reasons.

Deathtouch and lifelink are just keywords, and Judith, Carnage Connoisseur can apply them to any spell, as the ability says. However, those keywords only apply when something deals damage.

First reason: Disfigure gives a creature -2/-2, and that's it. It does not put -1/-1 counters on the target, and it does not deal damage. Neither deathtouch nor lifelink will apply here.

Second reason: Even as a 1/1, your friend's indestructible creature cannot be destroyed. Since deathtouch has the effect of "whenever this deals damage to a creature, destroy that creature," indestructible nullifies it.

To deal with an indestructible creature, you generally need to exile it, reduce its toughness to 0 or less (at which point it will die without being destroyed), or force the opponent to sacrifice it. Something like Disfigure is usually ideal for the job—it just wasn't big enough in this case. Grasp of Darkness would have worked perfectly.

Usually, Judith, Carnage Connoisseur wants damage spells like Lightning Bolt for removal, though those will have issues with indestructible. On Disfigure, you'd just use the Imp mode of her ability.