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.