Thinking of physical class in school, a liquid only can evaporate if the air is warm enough - and not saturated.
Sort of. Evaporation happens also at low temperatures; it's just a slower process. But the degree of saturation in the gas/air on top of the material (not necessarily a liquid) does indeed moderate the rate of evaporation also (i.e. vapor pressure). It's for this reason that on a dry, cold winter's day, ice can actually evaporate - a process called 'sublimation' (which also happens to some other solids; iodine comes to mind).
Taking this back to lubricants:
* Temperature is an important parameter like in all chemical reactions. The hotter it is, the faster it goes.
* W.r.t. RH I doubt it does much. Maybe very high RH (>90%) would slow down any evaporation, but this evidently has significant drawbacks and should be avoided.
* Evaporation of greasy lubricants will happen; the rate of this is dictated primarily by the chemical makeup of the lubricant. Generally, it's only the more volatile parts of the compound product that will separate out and/or evaporate.
I couldn't say whether the difference between 10% and 35% RH would be very significant in this context. I guess it depends on the greases involved and their chemical makeup. Even so, I expect the influence to be minor.
Avoid high temperatures; volatile parts of a lubricant can more easily spread out and wreak havoc this way. I guess we've all seen this happen on old lenses that have gone foggy. Leaving the camera bag in a sunny spot or for extended periods in a hot car would certainly accelerate this.