About the vapor barrier under the carpet...
What happens in warm weather is the moist air goes to the lowest place in the house (the basement) and can condense on cool surfaces, like basement floors. It might not appear as being wet, but the moisture is there. Then there is moisture vapor that comes up through the concrete.
What happens when there is carpet and underpad (without a vapor barrier) on the concrete is that over time it will get moldy underneath. It might take many years for it to get bad, but it will happen. The mold is actually eating the underpad and carpet and the spores that it releases are very bad for you. It will make your HT smell like a dirty gym.
Anything that touches concrete walls or floor needs to have a vapor barrier between it and the floor/wall or has to be moisture/rot resistant (ie: pressure treated wood). That includes your wood framing, wood flooring, carpet and laminate flooring. That is written in building codes everywhere.
It's actually best to have insulation, sleepers with sill gasket under them, then vapor barrier then subfloor (insulation on the cold side), but that can increase your build costs quite a bit.