I was wondering if anyone here feels it may be at all possible to develop a warp-capable, single-pilot fighter craft within the bounds of the Star Trek Universe.
Yes, it is possible but you'll get different results depending on the TV series/movies.
What would it take to make the smallest craft possible, but which was also capable of high warp speeds while carrying a modest compliment of weaponry with which to do battle?
For TOS and the TOS movies you're probably looking at a small ship (Bird of Prey-size or like the overloaded Orion ship from "Journey to Babel"). Small shuttlecraft type fighters would likely be picked off before they got close enough to do any serious damage.
In TNG, DS9 and later, power systems have gotten small enough that you could carry enough power for shielding on a small fighter to get close enough to do damage in numbers. A modest-sized ship like the Defiant was able to mount tremendous firepower but its design kept it from having too high a warp speed.
I'm thinking of a craft, roughly the size of a Formula One race car, capable of keeping up with & attacking a massive starship while travelling at high warp. How possible would it be to develop a warp engine system capable of generating the necessary output, while still being compact enough for a one-man fighter?
It's probably more likely for a one-man fighter in TNG/DS9/Voyager series. The main thing is whether the defending ship is able to outrun/maintain distance so it has time to pick off the fighters or if the fighters are able to overwhelm the defending ship.