The Project Image spell allows the caster to control an illusion of themselves from a significant distance away. They can see and hear through the eyes and ears of the illusion. The illusion can gesture and speak.
Can the caster cast spells (without material components) originating from the illusion? (through gesturing and speaking with the illusion)
Mislead has a similar effect, so the answer will likely apply to that as well.
In order for us to have a common language with which to determine how things work, we must stick to the idea that spells only do what they say and neither Mislead or Project Image offer an ability to cast through the illusion.
In fact, if you try to cast with Mislead:
the invisibility ends if you attack or cast a spell.
The spells dictate what you can and can't do and how they work, and neither have language regarding the ability to cast a spell through them.
I think you may be reading more into the following line:
make it gesture, speak, and behave in whatever way you choose
This does not mean you can cast a spell, it only means what it says in that you can have it do any of those communicative actions.
Just being able to gesture and speak does not give something a Spellcasting ability. Only having a spellcasting ability will do that, and without language stating such, the illusion does not have that ability.
As David Coffron points out, the Trickery Domain Cleric does have language in their Channel Divinity Invoke Duplicity(PHB, 63):
For the duration, you can cast spells as though you were in the illusion’s space, but you must use your own senses.
Spells only do what they say, so you cannot cast spells from the illusion's space.
Compare it to the Trickery Domain's Channel Divinity class feature:
as an action, you create a perfect illusion of yourself that lasts for 1 minute...
For the duration, you can cast spells as though you were in the illusion's space, but you must use your own senses.
This is the language that would be used if you had the ability to cast spells via the illusion of project image or mislead.
I mean, if the only requirement of the spell is that you need to be able to see your target and the range is permitting I don't see why you couldn't cast spells that way, using your project images vision. However, it wouldn't exactly be coming through the illusion, though you may be able to make it appear that way.
You can see through its eyes and hear through its ears as if you were in its space...
Unless otherwise explicitly stated, any spell you cast uses your actual body as the range-finder and origins for the spell. However, if your real body is hidden somewhere within range of your illusion's target, then in much the same way you can use your familiar's sight to target a spell if you're blinded, you can do the same with project image under this reading.
Meteor Swarm
At a range of 1 mile, you can follow a target and project image to catch up to it, allowing you to actually see it to make sure you hit the few squares out of a mile it's occupying.
Ravenous Void
This only has a range of 1000 feet compared to Meteor Swarm's mile, but 1000 feet of distance is plenty for safety in most situations, and can be achieved and maintained with dimension door. This will let you use your image to lure your targets in, then deliver a black hole on top of their heads without getting trapped in it yourself.
Mirage Arcane
This spell's target is up to 1 square mile of terrain, and its range is sight... it doesn't matter that the spell is coming from the original body, because it can reach all the way out to anything you see. Unless your target can teleport, you can literally shape the terrain to trap your target in a disadvantageous situation (1 square mile of lava or water can ruin the day for a lot of things), and disbelief will not help them, b/c the physical interaction remains.
