The first is how to safeguard its members from assassination.
The run-up to today's handover of power by Washington has seen a cascade of attacks on Iraqis cooperating with the new government and of direct threats against top leaders.
One group of insurgents released a recorded threat against Prime Minister Ayad Allawi in mid-June. In it, the group vows to murder Allawi the same way it disposed in May of Abd al-Zahra Uthman Muhammad (a.k.a. Izz al-Din Salim) -- a member of the former U.S. appointed Iraqi Governing Council -- in a suicide bomb attack.
"As for you, Allawi -- sorry, the democratically elected prime minister -- we have found for you a useful poison and a sure sword...we will pursue our mission [to kill you] to the end," the speaker says. "We will not get bored until we make you drink from the same glass that Izzedin Salim tasted. You have been spared from so many traps that we have set for you. You are the symbol of evil and an agent of infidelity. You are a hypocrite."
The voice is believed to be that of Abu Musab al-Zarqawi, a Jordanian militant alleged to have links with Al-Qaeda. His group and multiple other insurgent organizations show no sign of ending their resistance to efforts to build a new post-Saddam order.
But many analysts say equally tough challenges for Iraq's new government may come from within its own ranks. That is because the interim administration comprises representatives of factions that over the past months have acted as rivals as often as they have as partners.
Until now, the parties' experience working together has taken place in a setting largely controlled by the U.S.-led Coalition Provisional Authority (CPA). But as of today, the CPA is disbanded and the Iraqi leaders will be on their own in facing a host of continuing political disputes.
Abdel Saheb Hakim, an Iraq expert and human rights activist in London, notes that Iraq has no precedent for power-sharing in its political history to guide the interim government.
"There is no such experience in Iraq's history. In the past, all the governments have been dictatorships and the Iraqi people have no [prior] experience with democracy except a very little primitive democracy during the royal era,” immediately after Iraq gained independence under King Faisal II, Hakim said.
The government comprises representatives of Iraq's majority Shi'a Arab and minority Sunni Arab and Iraqi-Kurd communities. Since Iraq's independence in 1932, its governments have been dominated by the Sunni Arabs to the resentment of the other groups. The current government has a Sunni Arab president, Kurdish and Shi'a vice presidents, and a Shi'a prime minister in an effort to balance power.
In an effort to ensure rivalries within the government remain peaceful, all participating parties agreed in early June to disband their militias. Prime Minister Allawi said that their members will be integrated into Iraq's security forces.
"The vast majority of such forces in Iraq -- about 100,000 armed individuals -- will enter either civilian life or one of the state security services, such as the Iraqi armed forces, the Iraqi police service, or the internal security services of the Kurdish regional government," Allawi said.
But the groups have yet to be incorporated into national security forces, and it remains uncertain how they would react in serious power disputes. Overall responsibility for Iraq's security and stability is in the hands of the U.S.-led multinational force.
The seriousness of the power-sharing challenge was underlined in Kurdish unhappiness with the UN's resolution on 8 June endorsing Iraqi sovereignty. The resolution failed to include a guarantee sought by the Kurds that will maintain their present level of autonomy.
Some Kurdish parties have suggested they reserve the right to not cooperate with any Iraqi government that seeks to reduce their level of autonomy. The present level is endorsed by the Transitional Administrative Law adopted by Iraqi leaders under the CPA. But final resolution of the issue has been left for the writing of Iraq's constitution in 2005.
Faraj al-Haydari, an official of the Kurdistan Democratic Party (KDP), put the Kurdish position this way in an interview with RFE/RL in Baghdad last week: "There is an administrative law. This law should be implemented. If it is implemented, we will have achieved what we want. If it is not implemented, if it is canceled, we will take different measures than we are taking now."
With all participants in the interim government intent on ensuring they do not lose ground in the run-up to Iraq's first round of elections in January, the political jockeying in the months ahead could be intense.
The January elections will choose a transitional national assembly, which in turn will select a transitional administration to lead the country to direct election of a representative government by the end of 2005.
But how patient the Iraqi public will be with political infighting in Baghdad remains to be seen.
Al-Hakim says that today, Iraqis' greatest concern is security: "The Iraqi people now are concerned about their security, which is the first priority, more than the political situation. As far as the cabinet [of Allawi is concerned], I can't say anything [to praise it] unless this government achieves the most important goal of the Iraqi people, which is security."
The deep desire for order is the result of more than a year of insurgent suicide and car bombings, plus widespread lawlessness, including kidnapping for ransom. The instability has frightened the public and hindered reconstruction efforts that could generate jobs and lower the country's near-50 percent unemployment rate.
Some analysts warn that if the insecurity continues, new populist leaders could rise on promises to secure local areas by paramilitary means. If so, these leaders would be outside the U.S.-approved political establishment and could pose still more challenges to central authority.
But both Iraqi and U.S. officials are counting on today's handover of power to build public confidence in the new government and more representative ones in 2005. They hope that will embolden ordinary citizens to identify insurgents to security forces and enable them to finally crush them.
(The next part in this five-part series on sovereignty looks at the challenges for the Iraqi police.)
Click here for complete coverage and analysis of events in Iraq at RFE/RL's dedicated "The New Iraq" webpage.