What is the farthest object which we can get a direct Detailed visual image of using visible light which appears more than just a dot and falls into one of the following categories:
I think Pluto is the farthest we've imaged visually using New Horizons.
Can the Hubble telescope take detailed images of, say, a star ?
To address your last point, there are several stars of which we have been able to resolve images i.e. see the star as more than just a featureless point. There is a list of these stars on Wikipedia (I love that they put the Sun at the top of the list - true but pedantic :-).
The farthest away of the stars in the list is Epsilon Aurigae at about 2000 light years, so this probably answers the main point in your question.
However there is some ambiguity in your phrase direct visual image. We can detect supernovae in distant galaxies, though they cannot be resolved and appear as a featureless point. I'm guessing you mean to exclude objects like this, in which case Epsilon Aurigae holds the crown.
I'll add a theoretical limit to the actual record put forward by John Rennie.
To image an object as more than a featureless "point source", it must be resolved by the telescope. The angular resolution $\theta$ of a telescope is:
$$\theta\sim1.22\frac{\lambda}{D_{\rm aperture}}$$
$\lambda$ is the wavelength of light, $D_{\rm aperture}$ is the diameter of the telescope. Smaller angular resolution is better. The angular size of a distant object is $\theta=\frac{L}{D}$, where $L$ is the size of the object (e.g. diameter) and $D$ is the distance. Putting this together and solving for distance:
$$D = \frac{LD_{\rm aperture}}{1.22\lambda}$$
The current largest optical telescopes are $\sim 10\,{\rm m}$ in diameter. For best results, we want to be observing at the blue end of the visible spectrum at about $400\,{\rm nm}$. This just leaves the size of the source. From your list, the largest object would be a star (conveniently also the brightest, so it's easier to see at large distances). The largest known star is UY Scuti at a whopping $1700\,{\rm R}_\odot = 1.2\times10^9\,{\rm km}$. This would give a maximum distance for current telescopes of $D=2600\,{\rm ly}$. This lines up nicely with John Rennie's figure. The maximum I give is to just barely resolve the star (e.g. 2 pixels), so a bit closer would give a properly resolved image. To do better requires a bigger telescope, the next generation of optical telescopes will be $\sim 30\,{\rm m}$ in diameter, so they could get to triple the distance. You could also try to reduce the wavelength, but UV and X-ray observations must be done from space, so the telescopes are necessarily smaller and in the end you're better off from the ground in the optical.
If you stretch the definition of star a bit to include planetary nebulae (note: nothing to do with planets!), these have been observed out to $20\,{\rm Mpc}$, about $20\,000$ times more distant than Epsilon Aurigae. I wasn't able to confirm whether these were resolved observations, but their diameters are typically $\sim 1\,{\rm ly}$, which is about $10\,000$ times larger than UY Scuti, so resolving a big one at those distances is plausible.
