Brake fluids are either glycol-ether (Dot 3, 4 and 5.1) or silicone based (Dot 5 only). All glycol ether fluids are hygroscopic, which means they will absorb moisture from the air at normal humidity levels. There are only two ways to combat this effect: A completely sealed system, or regular fluid changes.
Ignore your fluids at your peril.
Most manufacturers list brake fluid change as a regular part of maintenance. For most cars it's based on mileage, but there's also a time limitation o when it should be done. For motorcycles, it's wise to do it yearly.
Being hygroscopic is also the reason you see the "Use only from a new, sealed container" warnings. Over time, it will naturally absorb moisture - water - and if you use old brake fluid, you're only putting moisture directly into your system. Since you can get brake fluid in very small packages, perform a change, and recycle the rest with your other oils.
The effect described in post 1 is normally the result of seal fatigue. The seals (called primary or master seals) in the master have failed, and won't hold the pressure anymore, as evidenced by years of correct operation and the fact that you can "pump them up". A simple rebuild (and at this point, rebuild the entire system) will resolve the problem.