First, it's going get get dumped the first time the "wrong" candidate wins the popular vote by 0.001% and some blue state has to vote all red or vice versa. Imagine all whining about the 'stolen' election in Florida, but an order of magnitude more annoying.
Maybe, but I doubt it. Under this system, the electoral college becomes a mere formality. People will of course be curious about how their state voted, but the determining factor is the popular vote, not the electors. It's a lot easier to justify "one person, one vote" than "one person, a variable number of votes according to a 250-year-old compromise that depends on your state's relative population."
Secondly, it's a huge incentive to cheat wildly in counting the votes. In order to prevent rampant cheating, you'd have to get all the States to agree on a single voting procedure and/or control of their election systems by the Federal government. If the latter's the case, you're right back to needing to amend the Constitution.
I don't follow. How is it more of an incentive to cheat wildly when you have to fake a 1-2% swing in 122 million votes nationwide compared to, say, the 5.5 million votes in Ohio?
Finally, there are plenty of States that aren't going to want this. If urbanization continues then a small number of urban centers will be setting policy for vast areas of the US about which they know little and care less. How many bitter gun-clinging, religious, 'fly over' states want to give over their power of self-determination to LA or NY?
By the same logic, right now we have rural areas disproportionately setting policy for urban areas. Under a popular vote plan, the rural areas would receive attention that more closely reflects their population. Is this a problem? Moreover, those states, and rural regions of those states, would still have disproportionate representation in the Senate and gerrymandered congressional seats: this proposal is only for presidential elections.
Also, I doubt the opposition would be that stiff in most states. There were only 19 states, worth only 189 electoral college votes, with a partisan advantage of more than 20 points in 2012 (i.e. more partisan than 60/40, ignoring 3rd parties). A national popular vote would allow the votes of the losing 40%+ in the other states and districts to still count.