When Igor (the big armor) is shown for the first time in the movie, the suit heads over to one of the crane's legs and stays under it to keep it from falling down but the crane doesn't fall down after all the suits have self destructed.
Brandt identifies herself as DHS when she attempts to arrest Tony Stark in the bar, however, the badge she displays and later strikes the Sheriff with is a Drug Enforcement Administration (DEA) Special Agent's badge. DEA is a Justice Department Agency not a Homeland Security Agency and their past and present badges do not look similar.
When Happy is in the ICU the nurse adjusts the roller clamp to the IV fluid. A pump controls the flow; adjusting the roller clamp would do nothing except shut off the medication, causing the machine to call for attention.
Hair lost or burnt off by those injected with Extremis grows back along with missing body parts. While hair is living within the follicle, once it's outside the skin, it's dead and cannot be regenerated in its current state. (As long as the follicle is not damaged, it can be regrown, but styling would not regenerate.)
When Tony is zip-tied to the bed, he asks one of the guards the distance between Miami and Tennessee. The guard says 832 miles, which is correct for some parts of Tennessee. This means Tony's armor is ~832 miles away. Even assuming his armor travels at 1000 miles/hour, it would take his armor almost an hour to get to him, rather than the few minutes shown in the movie. Tony's armor would have to travel at almost 10,000 miles/hour to get to Tony in 5 minutes. The current airspeed record for a plane is only 2,193 miles/hour. Tony's armor, at the current airspeed record, would still take 23 minutes to get to him.