Seems the no-see-um bite is pretty sharp.
No, as I lack insurance and my general experience with medical investigations are that they are incredibly inefficient if not totally dismissive of things like this.
Either the diptera involved is a no-see-um relative or the possibility of the "sting" sensation is actually coming from the ejection of foreign matter from under the skin instead of from a sting from the outside. This would seem more plausible given the black specks emerging from the skin and only rare indications of bites on the skin.