Thank you for the excellent question. Magnetism is a fascinating area of physics, which is complex and still being studied. The properties of a rare earth magnet are directly connected to its crystal structure [1,2]. To illustrate I will use the neodymium magnet, the general ideas apply to other rare earth magnets.

In order to flip magnetic spins in a neodymium magnet, the spin must change direction in the crystal structure. The crystal creates a very high energy barrier so the spin cannot flip unless a very strong magnetic field is applied.

Coercitivity of a magnet is the strength of magnetic field needed to demagnetize the magnet. It is measured in Oersteds (Oe). The coercitivity of neodymium magnets depends on the quality and structure of the crystal. It is about 10kOe [1]. 10 kOe corresponds to applying about a 1 Tesla field to the neodymium magnet. Applying a magnetic field smaller would induce almost no effect. The remnant magnetic field of a neodymium is about 1 Tesla. The magnetic field drops off very quickly, as one over distance cubed. Neodymium magnets placed 1 mm apart from one another would hold no effect. To place neodymium magnets into opposite polarity directly in contact with one another may have an effect but the mechanical strength needed to do this is not easily achievable in the lab.

A study was done in 2009 which showed if you heat a neodymium magnet and it has a poor crystal structure it can lose 10% of its magnetization over 30 years. As long as temperatures are kept at room level and the quality of the crystal is good, the loss of magnetization is negligible [2]. Neodymium magnets are highly corrosive and temperature dependent. It is more likely for it disintegrate or be heated before first principles cause the magnet to demagnetize.

[1] Herbst, J. F. "R 2 Fe 14 B materials: Intrinsic properties and technological aspects." Reviews of Modern Physics 63.4 (1991): 819.

[2] Haavisto, Minna, and Martti Paju. "Temperature stability and flux losses over time in sintered Nd–Fe–B permanent magnets." Magnetics, IEEE Transactions on 45.12 (2009): 5277-5280.

This is a challenging question, and I don't have a huge amount of experience in magnetism, but I will try to ballpark this answer. Neodymium magnets (rare earth) are incredibly resistant to losing their magnetic moment. Estimates put the time at around 10,000 years from what I have read. However, this is much shorter in contact with water, as neodymium corrodes. Also, high temperatures can reversibly or irreversibly cause loss in magnetic strength. In general, adverse fields (keeping the magnets repelling each other) will demagnetize weakly magnetic materials. However, neodymium magnets will rarely become demagnetized as they are strong.

Source for all of this: Alliance Industrial Magnets