First, I would like to clarify what is meant by "spontaneous". In chemistry, thermodynamically favorable reactions ($\Delta G < 0$) are indeed termed spontaneous because they will eventually occur, regardless of how slowly. But if transition states are unfavorable (high energy barrier), the reaction rate can be extremely slow --- millions of years in some cases. For example, combustion of wood is highly favorable and, chemically speaking, it is spontaneous. But that doesn't mean trees are unstable and likely to catch fire any minute.
Most biologists would probably call a reaction "spontaneous" only if it actually happens on a reasonable time scale, perhaps on the order of hours, depending on the context. So the terminology is a bit messy. But neither chemists nor biologists would call a molecule "unstable" if it has a reasonable half-life, I think.
Now for the actual questions. Dephosphorylations are typically not ATP-driven, because breaking the P-O bonds is usually favorable already (although I'm sure there exceptions). However, the reaction is typically too slow to be useful to cells, so it is not "spontaneous" in this sense, and therefore needs to be catalyzed by phosphatases. A typical example is alkaline phosphatase which dephosphorylates a wide range of proteins and other substrates.
Phosphorylation requires a phosphate source of course, but also an energy source, because forming P-O bonds is not a favorable reaction on its own (the reverse of the above). So adding a lone phosphate would not be favorable, but transfering one from ATP is much better, because this simultaneously involves breaking a P-O bond in ATP, "freeing up" energy to drive the reaction.
Finally, an interesting point is that ATP itself is actually a quite stable molecule, with a half-life of about two weeks. So ATP hydrolysis is not "spontaneous" in the biologist's sense, and enzyme catalysis is required (kinases). In fact, the combination of a high "energy content" and good stability is probably key to the role of ATP as an energy currency in cells: if ATP was indeed unstable, it would randomly phosphorylate everything. Stability means enzyme catalysis can be used to control phosphorylation events. Truly unstable compounds are never used by cells, because their reactivity causes damage and toxicity.
