Is it possible to visualise red shift? If a picture of a star or galaxy hurtling away from Earth is taken, does it appear red despite it being a different colour? Would a blue coloured star moving away from us appear red to us or vice versa? If so how do scientists understand if say, the red colour of a star is due to it having a cooler surface temperature (red supergiants like betelgeuse) or if it is due to the red shift?
 A: In order to get a visible (to the human eye) color change because of the Dopler redshift, an object has to:

*

*be moving at speeds at least ~5000km/s in regard to the human observer

*have a distinct color in the red-yellow-green range with rather high color saturation.

*be bright enough to be visible in color for the human eye

Less saturated color or a color of green-blue-violet range will need much higher speeds in order to generate a visible change.
The problem is, no handy objects with these properties exist.
Objects with 1000s km/s are not-so-much-nearby galaxies, they are white-ish for all practical purposes and they are faint enough so one needs a rather big telescope to see them (and good luck seeing them in color).
A: To detect the redshift of distant objects, we can use the fact that (to the best of our knowledge) the laws of physics are the same everywhere. This means the spectral lines of elements (both absorption and emission) will be the same at the location of the star as they are on earth.
We can measure absorption and emission spectra of elements here on earth and then compare them to the spectra we receive from distant objects. The amount the lines are shifted must then be due to the redshift (or blueshift if the object is relatively close and moving towards us).
You can see this visualised in this picture from Wikipedia
A: The standard view of General Relativity is that the locations of galaxies are actually fixed:  They don't change, except under the extremely long-term attraction of gravity (the weakest force, acting more imperceptibly than the weak nuclear, strong nuclear, and electromagnetic forces) between them, which is causing the "impending" collision of the Milky Way with Andromeda (that is due to be completed some billions of years from now).
What changes much more perceptibly is the wavelength of the light that stars emit, which shifts from the "hot" (blue) end of the spectrum toward the "cool" (red) end of the spectrum during the travel of the photons comprising it.  Even with that, there are subtleties of the spatial expansion which affect the size and exact shape of the "Hubble sphere", that is simply the region within which those changes in temperature can be astronomically observed, by us, as changes in the color of the light.
Because General Relativity is more complex than Special Relativity, the physicists Davis & Lineweaver have written a number of articles which each contain the phrase "expanding confusion".  Those articles try (verbally and diagrammatically, as well as mathematically) to sort out misunderstandings that inappropriately confuse Einstein's 1905 and 1915 theories.  The one that's easiest to read was in the March 2005 issue of the magazine "Scientific American", but a more elaborate version, containing diagrams at various levels of abstraction, can be found in Tamara Davis' doctoral thesis, freely visible at https://arxiv.org/abs/astro-ph/0402278.
Space itself does not consist of any mass, and spatial expansion is, as Davis very plausibly says, "not a force or drag" carrying any astronomical bodies with it:  Consequently, the strong possibility (implied by the homogeneity and isotropy which we observe) that their relative positions are changing with the expansion of space (first discovered through verification of General Relativity's predictions by Eddington's observations of curvature in the paths taken by starlight during the 1919 solar eclipse) is usually attributed to "dark energy" and "dark matter", although the shape of the universe (basically, the extent to which space, and especially time, are curved) is also a major factor in it.
As far as I know, the work of Davis and Lineweaver has never been challenged, although their attacks on the confusion of Special Relativity with General Relativity have been directed primarily at the pedagological (teaching) level.  Any citations to the contrary would be much appreciated.
A: 
If a picture of a star or galaxy hurtling away from Earth is taken, does it appear red despite it being a different colour? Would a blue coloured star moving away from us appear red to us or vice versa?

Red shift and blue shift is a terminology meaning shifting towards lower/higher frequencies. Usually one applies this terminology in visual spectrum, bounded by red and blue. Thus, the frequency of interest lies between these two and shifts in the direction of one of them - this does not necessarily involve noticeable change of the color.

If so how do scientists understand if say, the red colour of a star is due to it having a cooler surface temperature (red supergiants like betelgeuse) or if it is due to the red shift?

The temperature of stars is estimated by fitting their radiation spectrum with a black body curve, which has maximum at a certain frequency, proportional to temperature - hence different colors correspond to different frequencies, and blue is hotter than red.
