By Mike Collett-White and Asma Alsharif
CAIRO, July 6 (Reuters) - Egypt counted its dead on Saturday after Islamists enraged by the overthrow of President Mohamed Mursi took to the streets in an explosion of violence against what they denounced as a military coup.
At least 30 people died and more than 1,000 were wounded after Mursi's Muslim Brotherhood movement called "Friday of Rejection" protests across the country and tried to march on the military compound where the ousted president is held.
The most deadly clashes were in the Mediterranean city of Alexandria, where 14 people died and 200 were wounded. In central Cairo, pro- and anti-Mursi protesters fought pitched battles late into the night with stones, knives, petrol bombs and clubs as armoured personnel carriers rumbled among them.
It took hours to restore calm. The Nile River bridges around the landmark Egyptian Museum where the street fights raged were still covered with the debris of rocks and shattered glass on Saturday morning. Both pro- and anti-Mursi activists remained encamped in different squares in the capital.
The Health Ministry said 30 people were killed throughout Egypt on Friday, and 1,138 injured, state media reported.
State-owned newspapers said the army-backed authorities that took power on Wednesday and suspended the constitution, would announce the appointment of a prime minister on Saturday to run the country during a transition period.
Former U.N. nuclear agency chief Mohamed ElBaradei, 71, a leading liberal politician, was seen as the most likely candidate to lead an administration focused on reviving a shattered economy and restoring civil peace and security.
In an interview with Reuters, the country's main leftist leader, Hamdeen Sabahi, endorsed ElBaradei for the tough job, saying the transition should be short to amend the constitution and elect a new president and parliament..
The military has given few details and no timeframe for a new ballot - adding to political uncertainty at a time when many Egyptians fear violence could polarise society even further.
Egypt's first freely elected president was toppled after mass demonstrations against Muslim Brotherhood rule, the latest twist in a tumultuous two years since the fall of Hosni Mubarak in the Arab Spring uprisings that swept the region in 2011.
Five police officers were gunned down in separate incidents in the North Sinai town of El Arish, and while it was not clear whether the attacks were linked to Mursi's ouster, hardline Islamists there have warned they would fight back.
There more attacks on army checkpoints in the lawless Sinai peninsula overnight and gunmen fired on central security building in the town of El-Arish, security sources said.
A new Islamist group announced its formation in the lawless Sinai peninsula adjoining Israel and the Gaza Strip, calling the army's ousting of Mursi a declaration of war on their faith and threatening violence to impose Islamic law.
The group, calling itself Ansar al-Shariah in Egypt, said it would gather arms and start training its members, in a statement posted on an online forum for militants in the country's Sinai region recorded by SITE Monitoring.
The events of the last week have aroused concern among Egypt's allies in the West, including key donors the United States and the European Union, and in Israel, with which Egypt has had a U.S.-backed peace treaty since 1979.
Egyptian newspapers quoted ElBaradei as saying he expected Gulf Arab monarchies that were hostile to the Brotherhood's rule to pile in with financial support for the new authorities.
Only gas-rich Qatar provided substantial funds to Mursi's government with a total of $7 billion in loans and grants. Turkey and Libya also provided smaller loans and deposits.
Mursi's overthrow was greeted with wild scenes of celebration but infuriated supporters who fear a return to the suppression of Islamists they endured under military rule.
It has deepened Egypt's crisis. The Brotherhood has spurned army invitations to join an inclusive transition plan, culminating in fresh elections, saying it will not recognise the "usurper authorities".
Early on Friday, three protesters were shot dead outside the Republican Guard barracks where deposed Mursi is being held, security sources said.
The army denied responsibility for the shootings. An army spokesman said troops did not open fire on the demonstrators and soldiers used blank rounds and teargas to control the crowd.
It was unclear whether other security forces were present.
Later, tens of thousands of cheering Islamists gathered near a mosque in a Cairo suburb where they were addressed by Brotherhood leader Mohamed Badie, free to address them despite reports on Thursday that he had been arrested.
Badie, like some other leaders, pledge that it was worth "our lives" to restore Mursi to the presidency. But Brotherhood officials have also insisted they will not resort to violence.
After dark, running battles broke out in the area between Tahrir Square, scene of the demonstrations that toppled Mubarak, and the state broadcasting headquarters. Reuters journalists saw hundreds of youths from either side skirmish around the highway ramps of a major bridge over the Nile.
The violence will ring alarm bells in the United States. Washington has so far avoided referring to the army's removal of Mursi as a "coup", a word that under U.S. law would require a halt to its $1.5 billion in annual aid.
But many Egyptians see the army as a guarantor of stability at a dangerous time for the world's most populous Arab nation.
"Maybe they will need to issue a curfew. Maybe the trouble will last a few days," said Said Asr, 41, sitting with friends outside a Cairo cafe smoking a cigarette. "But the army is everything in this country. And they are taking control." (Reporting by Asma Alsharif, Mike Collett-White, Alexander Dziadosz, Maggie Fick, Alastair Macdonald, Sarah McFarlane, Shadia Nasralla, Tom Perry, Yasmine Saleh, Paul Taylor, and Patrick Werr in Cairo, Abdelrahman Youssef in Alexandria, Yursi Mohamed in Ismailia and Michelle Nichols in New York; Writing by Paul Taylor)