Where would we find the areas with a good house building boom? Which counties or cities care about quality work and are willing to pay for it? For one, Nashville and the more prosperous surrounding counties.
Is there a demand for decorative concrete at all? Somewhat.
Where could we hire a crew and not have to compete with so many illegal Hispanic crews? I'm not sure about this one these days. There has been an increased crackdown on illegals in the last year or two, but it can only deal with part of the problem. However, if you establish a reputation for good work done on time, you should be able to compete.
Are there any areas that have reputable builders? Parts of Nashville and other cities which have a higher income level and are willing to pay for a better quality product.
How are the county inspections? Inspectors will vary from location to location, from very good to poor. At least most of them are honest (in other words, you don't have to bribe them like in the northeast).
Are basements common? Is the work year round? Yes, basements are common and work is year-round (although most home builder try to "close in" before the start of the really cold weather). There probably wouldn't be too much residential concrete work from about mid December to mid February. There's some, of course, but cold weather is not a good time to place concrete as I'm sure you're aware.
It sounded like you were primarily asking about residential construction, so I've tried to answer your question as it relates to residential and not commercial construction.
By the way, I work for one of the largest structural engineering firms in the country. We don't do much work on residential so I only keep up with it in a general way.