What did they think about the size of the sun, it’s distance, why it existed, did they understand it gave off energy, etc?
You mean, that big ball of fire, pulled across the sky by a heavenly chariot?
Those are heathen myths, I’m talking about that ball who causes the days get shorter and shorter until December 21st, then the days get longer and we celebrate by having holidays around that time.
I think it’s pretty obvious to even the most primitive people that the sun gives off energy in the form of heat and light, and I think astronomy was pretty far along by the time people had any better concept of what energy was.
Astronomy is pretty old, how far back are you asking about?
Aristarchus made some reasonably good estimates (for the time) about the size of the sun and moon around 250 BC. I’d definitely consider him an astronomer, so you probably need to be more precise about what you consider to be “before astronomy”.
As late as the early 20th century, we really had no idea why the sun shines. The prevailing theory was it was slowly collapsing and the gravitational energy was producing the heat. That’s not unreasonable, but it would only allow the sun to keep shining for a few million years, which was therefore the best estimate as to its age by astronomers. Geologists on the other hand were seeing clear evidence that the Earth and its life were billions of years old. It was a major dilemma until radioactivity was discovered, providing clues to the mechanism by which it actually produces energy.
–Mark
Astronomical knowledge is a major part of Australian Aboriginal knowledge systems with indications it goes back a very long way. A great deal of research into this area is going on at the moment:
There are many people on earth today who have superstitious beliefs on the origin of the sun - plenty who believe that their god created it, so I am not sure when ‘before astronomy’ means.
My area of academic research is into non-literate knowledge systems - the way indigenous cultures manage to memorise vast amounts of practical information. All cultures I have studied have astronomical knowledge as the key aspect to maintaining a calendar for both resource management (be it hunting and gathering or farming) and for the ceremonial cycle. This information is stored in mythology - but that doesn’t mean the mythology is believed literally. That mythology will vary hugely between cultures. It is really important not to interpret indigenous stories as you would modern religions. The knowledge systems are integrated - spiritual with mundane - I can waffle on and on and on about this stuff so shall practice some restraint! 
In my field (primary orality), mythology along with song, dance and story, are mnemonic technologies, methods by which knowledge is made more memorable when it can’t be written down. The stories we are told are those told to children and form the basis of the system - something like a database structure. Further knowledge - more songs and stories - are added as the person gains higher levels of initiation, but these are restricted and will not appear in the collections of indigenous stories we non-initiated will hear. Restricting information is partly to do with power but also to do with avoiding the corruption of critical knowledge due to the ‘Chinese whispers’ effect.
The ability of Australian Aboriginal people to recognise and name star patterns, sun movement, moon patterns and so on, was noted as being far in advance of most non-Aboriginal people when first recorded.
The topics you ask about - how far away, the science we now know, obviously was not known. But then again, nor would it be a priority. The level of observation of the various bodies in the sky and ability to predict the movement over various cycles by the elders, was more advanced than I can do now, and more advanced than any but keen sky observers today.
The American Pueblo Sun-watchers were superb astronomers in terms of knowing the night skies and maintaining detailed observations. Such specialists could be found in indigenous cultures the world over.
So size, distance, why it gives off energy are recent. Most stuff that could be observed without telescopes and probes has been known for tens of thousands of years.
I think the sun is about as big as a quarter.
Some primitive folks thought the Moon was pretty small too: http://i.ytimg.com/vi/F4LwQhsFPlI/maxresdefault.jpg 
In particular, exposure to the sun “feels like” exposure to fire in a lot of ways. You can see by it, can feel the warmth on whatever side is facing it, and if you get too close (to a fire) or too much of it (for the sun) your skin turns red and hurts. Even absent clearly formed concepts of what “heat” and “light” are, and culture with experience with fire would naturally think that the sun is just a really big fire.
What’s always been interesting to me is the cultures who DIDN’T think it was just a big fire. They had stories that it was a chariot, or a god, or an eagle, etc. when it’s obvious at least that it’s a perfectly circular object. I’ve never understood how they reconciled that.
–Mark
FWIW the current belief among the flat earth fraternity is that the sun is 4000 miles away, and is the same size as the moon.
I think the problem is that you are taking one story literally and as the only explanation of the sun. I have found no indigenous culture which does that.
In small scale non-literate cultures, the area of my research, such as Australian Aboriginal cultures, there are no gods nor worship. The stories are part of a much more complex system. They use the terms ‘the stories we tell’ rather than fundamentalist type belief. I specialise in understanding the practical and scientific (they overlap!) aspects of the knowledge system.
Understanding of the sun will come through the interpretation of a whole range of stories at once. So the chariot type concept may be part of recording the movement and timing of the sun, but other stories may make reference to its roundness and heat.