Is there a type of sound within our visual spectrum that we can see with our eyes?
What we perceive as "sound" are (mechanical) oscillations of molecules from the source to the ear. This is for example why you cannot hear anything in vacuum, because there is no matter to oscillate. Light on the other hand is an electromagnetic wave. Therefore, there cannot be sound in our visual spectrum. It's in the wrong category.
However, if you want to ask "are there sound waves that you can perceive with your eye". In principle: if the sound pressure is high enough, you should be able to "feel" the pressure osscilations in the eye (and anywhere else in your body), so your brain would notice the sound even if you couldn't hear it, but still you can't see it.
If instead you ask yet another question, namely whether you can hear a sound and also believe that you have seen some light alongside, so that it seems that you have "seen" the sound, then you are no longer in the realm of pure physics: this is a known information processing error in many brains, where sometimes accustic signals are interpreted by the visual cortex. This is also known (especially in its more extreme forms) as synesthesia, but clearly belongs to biology and neuroscience.
No, there is not. The eyes are receptors of electromagnetic waves and therefore they don't percipe sound.
However, there are cases when you actually can see a sound effect on your own eyeballs, but they are unusual and a bit crazy. E.g. if you play low frequencies on a trombone and watch a screen with some repetition rate, then you can sometimes see an aliasing patterns due to the vibrations of your own eyes.
In principle, yes.
Sound waves are compressions in a medium, which in principle can be seen if the density contrast between wave crests and troughs is large enough, and the wave speed is small enough. In "everyday object", such as the air, neither of these conditions are fulfilled.
But one example of visible sound waves are the so-called baryonic acoustic oscillations, which are sound waves created in the plasma the first few $10^5$ years after Big Bang, with speeds of roughly half the speed of light. After decoupling, these sound waves got "frozen" in space. Much later, galaxies started to form and today, we actually see the density of galaxies increase and decrease on scales of roughly half a billion lightyears, which is the wavelength of the sound waves.
Yes. High intensity low frequency sound makes your eyeballs vibrate in your head, so you really can see (and feel) it. On a related topic, you can find the damped resonant frequency of your eyeballs by looking at the sweep line on an oscilloscope whose timebase has been set to around 50mS. If you then hit yourself downwards on the top of your head you will see a damped sine wave on the screen as your eyeballs vibrate.