What exactly do hunter-gatherers eat?
(and what about cooking?)

Because hunter-gatherers have lived in all kinds of environments, from deserts, to high mountains, to the arctic, to rainforests, to prairies, and to tropical islands, they have necessarily had to adjust their diet to the available foods around them.  They hunt and fish and collect foods that they learn, over thousands of years, how to find and prepare and eat.
Generally the more tropical the climate, the more they rely on plant matter for food, like fruit and roots.  Animal foods like fish and game are the major energy and nutritional sources for people closer to the poles, and in some cases plants contribute very little to the overall caloric intake.  
But all hunter-gatherers eat some meat or fish, regardless of where they live.
Cooking.  All hunter-gatherer groups cook their food, at least most of it.  Fruits and some organ meats and sometimes fish and grubs are eaten without being cooked, but most foods need cooking.  In fact, more and more anthropologists are realizing that cooking is one of the major factors in making us human.  While evidence of cooking before a few hundred thousand years ago is difficult to prove, it is becoming clearer that man was cooking at the time of the great changes taking place as man evolved from small  apelike Australopithecine species to the more manlike Homo erectus.  It allowed for a smaller gut, greater energy expenditure, more leisure time, smaller teeth, all of which helped allow the expansion in brain size, and led all of this evolution to modern man.
Cooking allows food to give up its nutrients more efficiently and allows us to eat far more quickly.  Early man would not have been able to eat the great amount of meats that he ate without cooking, as eating raw meat, for early man and us, is a tedious chore and we don't gain as much of the energy and protein from it if we eat it raw as when we cook it before eating.
Many root vegetables and other plant foods are more nutritious cooked than when eaten raw.  In fact, there appears to be no cases of long-term survival eating only raw food in the wild. 
It is not healthy to subsiste only on raw foods.  Some modern raw foodists get along, but only because the foods they eat are domesticated and often somewhat processed such as sprouted grains..  Even then, they suffer from dietary problems.
In short, early human groups and hunter-gatherers up to the present simply could not live eating wild foods without cooking.  (good books for reference:  Catching Fire:  How Cooking Made Us Human, and Evolution of the Human Diet).
Because of the great variety of environments and available foods of hunter-gatherers, each group can be studied separately as to what they ate, how the foods were collected, and how they were prepared.  In general, though, it is the men who do the hunting of game and fishing, and the women who gather the plant foods.  Grubs and other insects and worms are also eaten.
Some have said that the typical equality of women found in h-g groups owes to the fact that women actually bring to the group more food and more colories than the men in most h-g societies.  There is great importance attached to hunting, and meat is desired by all and is important for general nutrition, but it seems that the men only infrequently, maybe once or twice in a week, go hunting, while women forage daily and do the cooking. 
When men come back from a hunt, though, there is a lot of excitement, but it is curious that the person who actually brought down the animal that is providing the meat is not praised, as this could cause resentment by the other men.  In the Kung! society, it is the person who crafted the poison arrow that is praised, and he decides who gets which parts of the meat.  But everyone shares as equally as possible, again for decreasing any resentment among the group.  Sharing is a very important part of all h-g groups.

Hunter-Gatherers and Human Health. 
The fundamental truth:  our biology was meant for a different era and lifestyle.

Hunter-Gatherers rarely suffer from many of the chronic diseases that modern people so often suffer from. We are typically eating foods that our bodies never evolved to eat. 
"It is valid to view residents of affluent industrialized nations - be they Afican, Asian, Latin American, or Euro-American - as hunters and gatherers displaced over time to an environment much more foreign, even hsotile, to basic human biology" (from the Cambridge Encyclopedia of Hunters and Gatherers).
Our ancestors simply did not have access to processed goods that derive from grains,  and grains are not palatable or worth the while unless they are processed.  Sugar, lots of fats and oils, and dairy were not available either.  Even the meats from the game they hunted were lean.  Other differences that may affect differences in health is the relative rarity of alcohol consumption and the greater expenditure of energy through physical work of hunter-gatherers, although many in modern culture expend even more energy than hunter-gatherers and still succumb to modern degenerative diseases.
So here we are, living the post agricultural revolution culture, eating foods that are foreign to our bodies, and we end up with health problems. 
Some of the problems we have that hunter-gatherers are generally not afflicted with are:
1.  Diabetes
2.  Hypertension
3.  Dental Decay and Periodontal Disease (caused by sugar and refined carbs)
4.  High Blood Pressure
5.  High serum cholesterol levels (even though h-g'ers consume a lot of dietary cholesterol.
6.  artherosclerosis and coronary heart disease/heart attack
7.  obesity
8.  some cancers, such as lung (probably because of smoking in modern societies), colon (because of high fat consumption and low fiber in the diet), breast cancer (far less for h-g women).
9.  Communicative Diseases (contact with the outside world was far rarer for h-ger's, until recently, keeping these diseases much less common).

What health problems affect hunter-gatherers?

Hunter-gatheres typically can live longs lives, but many have their lives cut shorter because of certain problems, such as childbirth problems, traumatic injuries, long exposure to harsh environments, and many genetic or other kinds of diseases for which they have had no cure, such as appendicitis.  Though less common, viral and bacterial diseases can be devastating to a population because they have no adequate way to combat the spread throughout the population.
Hunter-gatherers suffer from a lot of health conditions that may not have caused problems in the past, mostly because of more contact with the outside world and the impact of industrial and agricultural societies on the ecological health of their territories. 
Many hunter-gatherers are still basically living their traditional lifestyles, but now also utilize imported substances like alcohol, tobacco, sugar, cooking oils, and even guns, etc, with predictable results.  Contact with more and more outsiders brings many communicable diseases for which they have built up little resistance.  Their foraging areas have been confiscated by outside farming communities, their water supplies polluted by nearby towns, and ancient migrations of game disrupted by development nearby.  It is hard to see how their societies can continue in a sustainable and healthy way for many more years.