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I found around a couple of damaged "gold coated" probes, after checking the inner wire it happens to be made of cooper, my questions here are simple
1) Why coat the tips if the rest of the probe is just copper?
2) In order to decrease the total resistance of the probes shouldn’t the entire wire be made of gold? (Common sense says that’s a bit expensive)
3) How can a simple gold coat help with measurements?
They are gold-plated because gold tarnishes far, far less readily than copper. Its purpose is not to provide a low-resistance conductor but to prevent tarnishing of the probe surface from affecting the measurement.
A thin plating of Gold is used in connectors because it doesn't corrode, so the connection will be more reliable.
It is not as conductive as copper, just a bit more than aluminium. Wikipedia has a table which says Silver: 63, Copper: 59, Gold, 41, Aluminium 35 (MS/m). So it might work for the leads, but would certainly be expensive, maybe a thousand dollars for a pair.
I've never seen gold plating on the tips of probe leads, on a lot of professional equipment. I think you're expected to press hard enough to make contact.
Wikipedia has more about the types of gold plating, the different thicknesses and alloys used.
1) Why coat the tips if the rest of the probe is just copper?
Many good conductors, such as copper, silver, and aluminum, oxidize at their surface when exposed to oxygen. This oxide layer has enough resistance that probe readings with light pressure on the conductor to be measured can be incorrect.
Gold doesn't form a high resistance oxide layer when exposed to air.
2) In order to decrease the total resistance of the probes shouldn’t the entire wire be made of gold? (Common sense says that’s a bit expensive)
Gold is slightly less conductive than copper and silver. The only reason to have gold coating is to improve the actual contact point.
3) How can a simple gold coat help with measurements?
Since there's no oxide layer naturally built up on the probe due to the gold coat, the probe requires little pressure and/or mechanical manipulation to take good, accurate readings from other conductors. Further, gold is a softer metal, and less likely to scratch the surface or conductor being probed.
That being said, there are many coatings that work well enough that gold coating provides little improvement over them.
Gold coated probes might have a slight advantage in situations where very small currents or resistances need to be measured and the resistance at the point of contact is a big factor. There might be some benefit to having gold coated probes for surfaces or conductors that cannot show marks and must not be scratched. There may even be some benefit when probing particularly hard or brittle surfaces where the softer gold may make slightly better electrical contact when the surface being measured won't deform to the probe tip. The gold coat is only a few atoms thick, though, so it wouldn't improve contact that much - perhaps enough in some extreme situations though.
While all of the OP's questions have been answered fully, a 'real world' example is something I learnt while I was doing work experience at an Aerospace/Defence company (many years ago at the tender age of 15):
For certain RF devices it is not uncommon for whole circuit boards to be fabricated with 'pure' (i.e. solid, not coated) gold tracks, pads, and vias on ceramic rather than fibreglass boards. In this sort of scenario, standard resistive, inductive and capacitive components are actually built into the PCB substrate rather than soldered on afterwards. For example, a resistor is created by filling the gap where the component would sit with a blob - literally - of resistive material, usually within a milled 'well' to stop it blobbing all over the place. This blob is then lazer-cut down to the correct, immaculately precise value. A resistance meter is of course required, which is where gold-plated tips are essential.
As the tracks themselves often have a critical resistance it is important not to damage them. Jabbing at a gold track with anything harder will disturb the track's surface and potentially comprimise it's resistance, hence the need for gold tips - gold-on-gold of will never cause this (unless you try really hard). For the trimming of some components the meter's tips may even be solid, softer gold because (a) the contact area for measuring is so small, and (b) there is a limit on the pressure that can be applied to the track/component; the softness of the probe-gold means the tip will likely break before any damage is done to expensive PCB.
