In short, size does not matter. Also, intelligence is not just one ability. Rather, there are different types. For example, there is spatial intelligence (how good we are at remembering where things are, moving things in our 'mind'), verbal intelligence, and emotion intelligence to name just a few.

There is also research that shows when we disconnect the tissue of the brain (which is called the corpus callosum) that connects the left and right hemispheres, individuals are able to function normally. This shows that we don't the two parts of the brain two works together all the time.

No.
Intelligence in animals is largely a function of the complexity and sophistication of the connections between sense organs and activity organs such as muscles. Having more nerve tissue allows you to have a more sophisticated algorithm, but does not imply it.In animals, the general rule for intelligence is the ratio of brain size to body size, since you need more nerve tissue to coordinate more body mass, but even so, this is hardly a definite algorithm. Squids and octopuses are far more intelligent than most herbivorous mammals, despite having substantially smaller brains, even relative to their body mass.

Nope--cows have bigger brains than we do, and they're not very smart. It's all about connections in the brain. You do need a certain minimum size of brain for particular tasks, which is why things like emotion, analysis, art, and mathematics. But it's mostly how complex the "wiring" is. Your brain makes new wiring for each thing you learn. Most of the wiring is your neurons (brain cells) making new connections to each other.

The question of to what extent brain size and intelligence are related is complicated. Brain material is very energetically demanding. In modern humans,the brain comprises only 2% of the total body weight, but consumes roughly 20%of the bodys oxygen and caloric budget. This high cost is justifiable as we use our brains to perform complex cognitive exercises. By comparing different species we can see that when the successful exploitation of the environmental requires higher brain functions, we see the evolution of larger brains. In bats, for instance, those species that live in forested areas, where there are more obstacles to avoid when flying, brains are typically larger than in those species that hunt in open areas. This is presumably as the brain needs to be able to process more sensory information in the more complex habitats. If you look at the brain size of the species that led up to modern day humans there isa clear pattern of increase over time. This would not have come about unless those individuals with larger brains were more successful (for instance at hunting, making tools, communicating, etc) than those with smaller brains. Large brain size, however doesnt tell us everything about intelligence. The species with the largest brain by volume isnt humans (average volume 1400cubic centimeters), but sperm whales (average volume 9000 cubic centimeters). However, due to their large size, sperm whale brains comprise only 0.02% of their total body weight. Large brain size also doesnt mean that a species will necessarily do better than species with a smaller brain. For instance,Neanderthals (a sister species to modern humans from the genus Homo), had on average larger brains than do anatomically modern humans, but still went extinct (there is, however, some debate amongst scientists whether some Neanderthals interbred with anatomically modern humans and thus disappeared though genetic absorption). Nevertheless, it would appear that, overall, brain size (as measured as a percentage of body weight) is related to the ability to perform complicated cognitive tasks.

Great question, but tough to answer. If we are comparing two individuals of the same species, then we need to think about what specific "parts" of the brain might be related to "being smarter." Also, it's difficult to define "smarter," so that makes it tough to get at this kind of question. Lots of people have asked this type of question though. In fact, Albert Einstein (who we all agree was "smart") donated his brain for such a study -his preserved brain was just recently re evaluated using new techniques and although his brain was not substantially "bigger" overall, there were parts that had slightly different structures and surface areas. The higher cognitive function areas were much more "developed" than a "typical" brain. But again, it can be difficult to know what "typical" means. So, although there are some studies out there, I'd have to say we don't really have an answer to your question except that brain weight is not correlated precisely with IQ (just one measure of "smartness, which is controversial, too).

But you can also think about your question comparing across species. Let's compare humans to a chimpanzee. Whatever distinguishes human primates from other animals must reside in the unique features of human development, especially in development of the brain. There are at least 5 features that distinguish human brain development from that of other primates:

1. Retention of fetal neuronal growth rate after birth.
2. Migration of cells from pro encephalon to diencephalons (you can look at a "brain atlas" to see these parts.
3. Activity of transcription (gene expression) - it's much higher in humans.
4. Presence of a specific form of the FOXP2 gene (critical for speech, language.
5. Continuation of brain maturation into adulthood.

This last one is the "size" aspect - human brains keep growing well into early adulthood while in chimps (for example) the brain hardly grows at all after birth.

I'll leave you with a quote that I love, basically asking "can my brain figure out its own mystery?"

"What is perhaps the most intriguing question of all is whether the brain is powerful enough to solve the problem of its own creation [during embryonic development through childhood]."Gregor Eichle (1992).