(Updates death toll, adds Arbil bombing)
By Ahmed Rasheed and Isabel Coles
BAGHDAD/ARBIL, Iraq, Aug 23 (Reuters) - Bombings across Iraq killed at least 35 people in attacks that appeared to be revenge for an assault on a Sunni mosque that has deepened sectarian conflict.
A bomb also exploded in the northern city of Arbil on Saturday, a rare attack unsettling the relative stability the capital of the semi-autonomous Kurdish region has enjoyed.
Local television footage showed firefighters dousing the charred remains of a car in Arbil. A Reuters journalist earlier saw a cloud of smoke, but the source was not clear.
In Baghdad, a bomber rammed a vehicle into an intelligence headquarters, killing at least eight people, police and medical sources said. Near Tikrit, a suicide bomber driving a military Humvee packed with explosives attacked a gathering of soldiers and Shi'ite militias overnight, killing nine.
Shi'ite militiamen machinegunned 68 worshipers at a village mosque in Diyala Province on Friday as politicians try to form a power-sharing government capable of countering Islamic State militants.
An advance by Islamic State through northern Iraq has alarmed the Baghdad government and its Western allies and drawn U.S. airstrikes in Iraq for the first time since the withdrawal of American troops in 2011.
Although the air campaign has caused a few setbacks for Islamic State, they do not address the far broader problem of sectarian warfare which the group has fueled with attacks on Shi'ites.
Bombings, kidnappings and execution-style shootings occur almost daily, echoing the dark days of 2006-2007, the peak of a sectarian civil war.
In addition to the Arbil attack, three bombings that appeared to target Kurdish forces killed 18 people in the city of Kirkuk, 250 km (155 miles) north of Baghdad, security sources said.
Islamic State routed Kurdish forces in its latest advance through the north.
Two of Iraq's most influential Sunni politicians suspended participation in talks on forming a new government after the militiamen carried out the mosque attack.
Deputy Prime Minister Saleh Mutlaq and Parliament Speaker Salim al-Jibouri have pulled out of talks with the main Shi'ite alliance until the results of an investigation into the killings are announced.
Jibouri, a moderate Sunni, condemned both Islamic State as well as the Iranian-trained Shi'ite militias who Sunnis say kidnap and kill members of their sect with impunity.
"We will not allow them to exploit disturbed security in the country to undermine the political process. We believe the political process should move on," he told a news conference on Saturday.
Iraq's new Shi'ite prime minister, Haider al-Abadi, faces the task of trying to draw Sunnis into politics after they were sidelined by his predecessor Nuri al-Maliki.
Maliki stepped aside after pressure from Sunnis, Kurds, some fellow Shi'ites, Iran and the United States.
Iran, a regional power broker with deep influence in Iraq, is sending its foreign minister, Mohammad Javad Zarif, to Baghdad on Sunday for talks with Iraqi officials. (Additional reporting by Alexander Dziadosz; Writing by Michael Georgy; Editing by Rosalind Russell)