thermodynamics - How do warm-blooded animals keep their temperatures constant?

Since there seems to be several distinct sub-topics in your question, I will answer them one-by-one:

1). There are a variety of mechanisms that allow endothermic animals to maintain thermal homeostasis in a cold environment. The main ones are:

a). The shivering response: When the core body temperature of a endotherm drops below a critical value (36.8C in humans), it causes the posterior hypothalamus to stimulate certain skeletal muscle groups (especially around vital organs) to start to "shiver" rapidly, generating heat.

b). Compared to ectotherms, endotherms have more mitochondria per cell, thus allowing them to have a higher metabolism. Since metabolism always generates heat, an [general] increase in cellular metabolism will cause an increase in body heat.

c). Many endotherms have layers of insulating matter, such as fur, blubber, feathers…etc, allowing them to preserve body heat. In addition, endotherms can also route blood away from capillaries via vasoconstriction of arterioles, reducing the area in which heat can be lost.

d). As mentioned by Memming, brown adipose tissue also plays a role in temperature regulation. Thermoregulation-based metabolism in brown fat causes the P+ in the electron transport chain to go through thermogenin instead of ATP synthase. This process generates heat, but no ATP.

e). Some endotherms, such as penguins and arctic wolves have countercurrent exchange in their capillaries. This is when warm arterial blood "passes" some of its heat to cooler veinous blood. This feature allows some of the heat normally "wasted" into the air to be recycled back into the body.

Note: Though it is true that endotherms are able to keep their body temperature constant irrespective of their surroundings (ignoring extremes), they do this at a cost of requiring significant amounts sustenance. Most endotherms require much more sustenance than ectotherms.

2). Your second sub-question is very interesting. Though it is true that generating heat will require substantial amounts of carbohydrates/fats/...etc, it does not necessarily mean that the net consumption of sustenance in cold environments is greater than that in normal environments. In most cases, endotherms in cold environments will have exhibit significantly less activity than when in an optimal environment. The decrease in activity when in a cold environment will likely balance out the increase in thermo-regulation based metabolism. This likely explains why people tend to drink equal or slightly less amounts of water when in cold environments. One more thing: some reactions heat-generating reactions (like the alternate ETC pathway) do not require water. Glycolysis and Krebs actually generates water (not to say that there is a net gain in the body :))

3). In truth, nothing "prevents" ectotherms from generate heat. They simply do not have the cellular "machinery". Ectotherms metabolize in ways very similar to other organisms, using molecules like ATP, glucose, fat…etc. Unlike endotherms however, they spend very little of their energy on temperature regulation. As a consequence, their overall metabolic rates are dependent on the external temperature. The point is this: A substantial portion of endotherm sustenance is used to generate heat. Only a small (if any) portion of ectotherm sustenance is used to regulate heat. As a result, endotherms require much more nourishment than ectotherms.

4). I am not entirely certain what you mean by "evolutionary path", but I will just say this: In many ways, endotherms and ectotherms are organisms that have found different ways to the same problem; how to regulate body heat for maximal survival and reproduction. Basically,

ectotherm- more dependent on environmental temperature, requires less sustenance

endotherm- less dependent on environmental temperature, requires more sustenance

5). You are quite correct. The cellular and genetic components are very similar. Some morphological aspects seem to be shared as well.

Sources:

Cambell & Reece (2010) Biology (9th ed)

Swan, K. G.; R. E. Henshaw (March 1973), "Lumbar sympathectomy and
cold acclimatization by the arctic wolf", Analysis of Surgery 177 (3):
286–292,

Guyton & Hall (2006) Textbook of Medical Physiology. (11th ed)

Romanovsky AA. (2007). Thermoregulation: some concepts have changed.
Functional architecture of the thermoregulatory system. Am J Physiol
Regul Integr Comp Physiol. 292(1):R37-46.