When Harley first meets Tony in his garage, he realizes who Tony is after seeing the Mark 42 suit - he acknowledges this by handing Tony the newspaper with the article about his Malibu home being blown up. So why did Harley ask Tony his name a few minutes later ("Harley. And you're...?")?
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 and Rhodey are at the docks and taking cover, Tony asks for more ammo from his pistol, even though Rhodey tells him their bullets are not compatible, which Tony insists that they are. While Tony may not have any military training like Rhodey does, he is a notorious former weapons designer, and should know that certain pistols can only use certain caliber bullets.
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.