You need to process things in a timely manner, as Privateer said. Trying to pick things that are still processed up gives you information on when they will be ready, so you can check how long it's left (but you don't get that info by looking at them, so you have to go to the location).
It's not possible to warn you about rotting of meat (at least, and probably of furs as well) because there's a significant random factor involved, so each morning when the day ticks over there's a chance things will degrade, but there's also a chance it will not, so it's not known beforehand (You can see that clearly by having several different pieces of meat acquired at the same time that degrade at different rates. While cooking, for instance, left over partial pieces may degrade slower than the whole pieces it used to stack with, despite the stack being left in the cool cellar all the time, while the one used for cooking was kept in the inventory by the fire at least during the cooking process).
Also note that degradation happens in steps, so meat first goes stale, and fur/skin degrades one level. This means that by the time the skin rots it has already degraded in steps down to harsh.