How is energy not stored in bonds?
Let's start with the specific examples. For the conversion of ethene to ethane, you need a reaction partner to provide the extra hydrogen, e.g. dihydrogen:
$$\ce{H2C=CH2 + H2 -&gt; H3C-CH3}$$
In the reaction, a single bond and the pi-bond of the double bond are broken, and two single bonds are formed. If you look up the four bond dissociation energies (two broken and two formed), you get a fairly good estimate of the reaction enthalpy.
$$\ce{ADP + phosphate -&gt; ATP + H2O}$$
I'm leaving out the charges and the protonation state of phosphate to make things simpler. If we look a bit closer, we can write:
$$\ce{RO3P-OH + H-OPO3R' -&gt; RO3P-OPO3R' + H-OH}$$
So when ATP is formed, we are not only making the phospho-anhydride bond (and a bond within water), but also breaking two bonds.

Lastly, when teaching cellular respiration, we speak of energy being stored in the phosphate bonds when ADP is converted to ATP and in photosynthesis, of the energy from sunlight being stored in the bonds of glucose.

Is this line of thinking correct, if not why?


You could say the energy is stored in the reaction, and when the reaction runs in the other direction (hydrolysis), it gets released again. Similarly, glucose is a source of free energy only if oxygen is available as reaction partner. Breaking glucose into atoms is not a source of free energy.

Let's look at the first statement It takes energy to break bonds.

Is this true of spontaneous processes?


Here is the reaction this refers to:
$$\ce{molecule -&gt; free atoms}$$
The entropy of reaction is expected to be positive (more particles on product side). If the reaction were also exothermic, these molecules would not be stable under any condition (Gibbs energy of reaction would be negative at any temperature). So bonds of molecules that actually exist are expected to have a positive bond dissociation energy.
The other questions
The answers to the rest of the questions follow from the information above.