1. ATP hydrolysis can happen on its own. Every chemical reaction can happen on its own without a catalyst. Even endothermic reactions (reactions that are "non-spontaneous") can happen on their own if the particular conditions the reaction will take place in are favorable.
However, for a reaction to happen several things must occur:
1. the energy of the reactants must be of sufficient magnitude such that a collision between the reactants has enough energy to generate the unstable, high energy transition-state. The transition state is a fleeting entity, a combination of the reactants (in this case ATP + H2O) as they undergo some reaction. This is where enzymes come into play -- by lowering the activation energy; or the energy necessary to generate the transition state from the reactants. They do this by stabilizing the transition state, thus lowering its inherent energy, and making it easier to compose.
2. the reactants must be oriented in such a way that the necessary components of each reactant molecule to participate in the reaction can physically interact with one another. Enzymes can orient the molecules in such a way that the correct parts of each reactant can interact with one another. In the case of ATP hydrolysis, the Oxygen atom of water must be able to directly interact with a phosphorus atom in ATP. The respective enzyme will orient this interaction in the most efficient way possible.
As a quick aside the factors that contribute to the relatively large, negative change in free energy of ATP hydrolysis are:
1. the negative charge repulsion of the phosphate groups bound to each other
2. the greater resonance stabilization capabilities of Pi and ADP on their own as compared to ATP
3. stabilization due to hydration: more H2O molecules can interact with Pi and ADP on their own to stabilize them as opposed to the phosphoanhydride linkage of ATP.
So, we can see that ATP hydrolysis can happen on its own if a water molecule and ATP molecule with sufficient energy collide with one another in just the right way in that the oxygen is oriented in a precise manner which allows it to exchange electronic-density with the phosphorus (this is simplified, you can look up the mechanism of ATP hydrolysis). It comes down to statistics, how many of the H2O and ATP molecules in our body will collide in this precise manner without enzyme catalysis to initiate the reaction? Surely, some will due to the sheer number of molecules of each present in the body, but the proportion is rather low.
2. ATP hydrolysis (more specifically ATP decomposition, not necessarily always required a water molecule itself as the molecule used to break ATP into ADP + Pi) can take place anywhere inside the cell. As just discussed ATP hydrolysis "can" happen on its own. Aside from that, all that is necessary for ATP decomposition to occur very readily is an enzyme that can utilize ATP hydrolysis/decomposition to power some other cellular process, and the right cellular conditions that activate that enzyme. Consider enzymes such as hexokinase, citrate lyase, pyruvate carboxylase, and DNA polymerase.
Hexokinase is a key enzyme in gycolysis, utilizing ATP to phosphorylate glucose into glucose-6-phosphate, a key intermediate in glycolysis and the pentose phosphate pathway. Where does this happen? In the cytoplasm.
Citrate lyase, the enzyme that catalyzes the cleavage of citrate (used as a transport form to get OAA and acetyl CoA from the mitochondria into the cytoplasm) into acetyl-CoA and oxaloacetate utilizing ATP. This again, happens in the cytoplasm.
Pyruvate carboxylase, the enzyme which can generate oxaloacetate from pyruvate using ATP, is found in the mitochondria.
Lastly, DNA polymerases -- the wonderful enzyme family that allow for the replication of DNA -- must utilize the cleavage of a pyrophosphate from deoxynucleotides (either dATP, dCTP, dTTP, or dGTP) to add the respective base (either A, C, T, or G) to the growing DNA strand. Of course, this takes place in the nucleus.
3. As just discussed, ATPases, or enzymes that somehow utilize the cleavage of ATP (or some molecule bearing similar properties; thus it is better to think of all XTPases where X = G, C, A, or T; or even U; as having the potential to power cellular reaction). Yes, we typically think of the Na+/K+-ATPase (and other transmembrane ATPases) which utilizes the hydrolysis of ATP to pump 3Na+ out of and 2K+ into the cell, but in reality all of the enzymes just discussed above (and many, many more!) are ATPases in the sense that they break down ATP into ADP + Pi and use this energy to power another reaction.
4. ATP (adenosine TRIphosphate) can be completely hydrolyzed to AMP under a number of circumstances. However, when ATP is being used rapidly an enzyme called adenylate kinase can actually form more ATP from 2ADP in the reaction: ADP + ADP <--> ATP + AMP. The reaction of DNA polymerase, as an example, cleaves dATP directly to PPi + AMP (which is attached to the DNA strand).
Although I'm not complete sure why the cell does not use ADP as readily as an energy source, I would have to imagine the cleavage of an inorganic phosphate from ADP is simply less energetically favorable than is the cleavage of a phosphate off of ATP. You can look further into this by considering the factors that would come into play (charge distribution on ATP vs ADP, size, enzyme specificity, solvation factors, relative energy of the terminal phosphoanhydride linkages of ATP vs ADP, steric factors, etc.).