There are actually three parts to that question:
Conservatives are generally inclined to answer "yes" to all three questions. Their star witness is Lebanese Druze leader Walid Jumblatt--not exactly a paragon of the democratic spirit--whom the Washington Post's David Ignatius nevertheless quoted as saying,
It's strange for me to say it, but this process of change has started because of the American invasion of Iraq....I was cynical about Iraq. But when I saw the Iraqi people voting three weeks ago, 8 million of them, it was the start of a new Arab world.Interestingly, the left offers a fair bit of qualified agreement--including even some credit to the Bush doctrine and its application in Iraq. There is general admiration for the elections there and for the new post-Arafat order among the Palestinians; particular satisfaction over Mubarak's concessions in Egypt; and widespread skepticism only regarding prospects in Lebanon. (Particularly amusing is Flynt Leverett's New York Times Op-Ed warning about the possible disastrous consequences of Syria losing control of Lebanon. He warns that the US might have trouble "containing Hezbollah without on-the-ground Syrian management"--as if Syria weren't Hezbollah's primary lifeline.)
In fact, leftist skeptics are correct in not swallowing conservative triumphalism whole--democracy is far from a done deal in any of these countries. Their skepticism, however, isn't necessarily wisely distributed.
Possibly the most promising case is Lebanon, which was at least somewhat democratic before civil war broke out in 1976, and whose current dictatorial government is foreign-imposed. Its foreign occupier, Syria, is now straining under American and even some international pressure to leave Lebanon to the Lebanese. For Syria, the fall of Saddam Hussein meant the elimination of a major source of distraction from Syria's misdeeds, and of much-needed oil-smuggling revenue.
On the other hand, the Syrians have enormous incentive not to give up so easily--their economy is heavily dependent on its plunder of more prosperous Lebanon--and a long record of being as ruthless as they need to be. If Syria were to react to Lebanese rebellion not by withdrawing its troops, but rather by sending its tanks into the streets of Beirut, it's highly doubtful that there would be any significant Lebanese resistance, and foreign intervention--American or Israeli--is no more likely. Driving Syria out of Lebanon will require much more than a few civilian protests--sustained economic, political, and perhaps even military pressure may be necessary. Fortunately, Syria's government is in a very weak position, and Europe and America are unusually united in this effort.
The new government in Iraq has, in some ways, better prospects than the Lebanese democrats--after all, a supportive American occupation is much less of an obstacle than Syrian hegemony. But Iraq's population is at least as fragmented as Lebanon's, and has an even worse historical record of hostility. Moreover, Iraq's domineering neighbor, Iran, is as ruthless as Syria, and much more resilient to Western pressure. American troops can head off an actual invasion, and help suppress terrorist uprisings. But they can't prevent Iraq's government from disintegrating into ethnic, tribal or regional warfare, or falling into Iran's orbit quasi-voluntarily.
At least Iraq's democracy is off to a fairly promising start. Egypt's and Saudi Arabia's token gestures towards democracy are neither sincere nor likely to be fruitful. Neither country has anything like a viable democratic opposition, although both are threatened by thriving Islamist terrorist movements. Under the circumstances, democracy's poor prospects in these nations may not actually be an unalloyed misfortune--particularly for the US, but also for citizens of those countries, considering the precedent of Algeria's unfortunately premature democratic experiment in the early 1990s.
As for the Palestinians, their recent exercise in democracy spoke volumes. The new Palestinian Authority President, Mahmoud Abbas, actually has a remarkably good record as a supporter of peace and reconciliation with Israel. But Palestinian society is still clearly dominated by supporters of never-ending, all-out terrorism--and not, needless to say, by democratic reformers--and Abbas has no power even to take significant action against the terrorists, let alone to defeat them. Perhaps one day there will be enough popular support for peace with Israel to mount a credible opposition to the terrorist warlords who effectively rule the territories. But that day still looks very far off.