Quite the contrary, Iran does not have a democracy of any sort. Although the presidency is an elected position, the highest executive authority in the country is the Supreme Leader, who is chosen by a government-appointed deliberative body known as the Assembly of Experts. Even the pseudo democratic institutions in Iran are now rigged, with the majority of political parties banned from running and clear evidence of voter fraud in the most recent election.
The U.S. would be very foolish to invade Iran. The threat of nuclear proliferation would be their only real justification for any sort of military strike; this has to be measured against the adverse effects it would have on America's international image, not to mention the civilian deaths that would result from an attack. Regime change in Iran? That's bound to happen in the near future anyways. The government is fragmented and there is plenty of internal opposition to the Supreme Leader, not to mention the huge amount of discontent among the citizens in the country. Iran is a highly progressive country with a highly conservative government.