interesting post, and i have seen it before in other forms. but you are confusing a couple of points that need clarifying. unfortunately, some of the clarification gets technical, but if you are interested, read on.
there is no such thing as aerobic and anaerobic energy. there is aerobic and anaerobic respiration. you are partially right in that aerobic respiration takes place in the mitochondria, while anaerobic respiration can take place outside of the mitochondria.
ATP is the unit of energy that is used by the tissues. ATP produced by aerobic respiration is the same as ATP produced by anaerobic respiration. it is just that less ATP is produced by anaerobic than by aerobic, so the latter is preferred in terms of efficiency.
aerobic respiration is when ATP is generated by tranforming glucose through glycolysis into pyruvate, which then enters oxidative phosphorylation. this second step, which happens in the mitochondria, requires oxygen. in the end you are (essentially) left with carbon dioxide, water, and ATP (for fuel).
anearobic respiration is when ATP is generated by transforming glucose (again through glycolysis) into pyruvate. the next step doesn't require oxygen and pyruvate is converted to lactate by the enzyme lactate dehydrogenase. i was very careful about the last statement. it does not necessarily take this path because there is low oxygen...it more has to do with the overall availability of substrates for oxidative phosphorylation.
anaerobic respiration produces only 2-4 ATP (don't sweat the details on why i report a range) and aerobic respiration produces 36-38 ATP per glucose. thus the latter is more efficient, though the former is faster. they are usually both happening at the same time, in a steady state, but when the appropriate substrates are not there for aerobic respiration, or you need a quick energy boost of ATP, anaerobic respiration dominates.
if you want an explanation for why people get out of breath when they exercise, that will take another physiology lecture. but it IS related to aerobic and anaerobic respiration, and is mostly a function of the elimination of carbon dioxide (rather than delivery of oxygen). and, as i said earlier, carbon dioxide is a main byproduct of aerobic and anaerobic respiration and so your statement on breathing isn't really correct. in fact, the reason you are breathing harder IS NOT because you are overheating. unlike dogs, we sweat, and thus don't depend on panting for heat elimination. we breath harder to eliminate carbon dioxide.