I would like to make a composite whose end goal is micro-mesh-like with pretty much zero strength/stiffness requirements except to hold its own shape with confidence. 

No strict specifications to meet on its permeability, but roughly speaking it should allow air to pass through, but not splashes of water.

Just wondering, does anyone have any experience? Is it possible at all?

Naively speaking, I am thinking of sourcing some loosely woven fibres, brushed/sprayed with very thinned-down resin, such that there's just barely enough resin to attach fibres together, flowed around by capillary action before curing.

Would such an idea work? Are there more appropriate terminologies for anything I've described?

Its a difficult one because in all normal composites, you want full wet out of the fibres, which is the opposite of what you are after.  You could experiment with it but the less the fibre is wetted out, the weaker it will be but there are also issues with consistency between layers. 

Warren Penalver
Easy Composites / Carbon Mods - Technical Support Assistant

Thanks for your feedback Warren, it is indeed a very weird application. I have also been considering starting with a mesh instead of loose fibres (think something like a fly screen?), but still not quite sure how to deal with the resin parts.

Would you know of any particularly thin/flowy resins? Or if not, what would you recommend to thin out a

If your mesh structure is reasonably rigid enough to hold its shape, you could try making a resin bath and dip the mesh into it then lift out.  Trying to thin the resins down can cause a lot of issues itself. 

Warren Penalver
Easy Composites / Carbon Mods - Technical Support Assistant