* Gaza rocket fire tapers off under Egypt's mediation
* Hate crimes underscore stagnant peacemaking efforts (Releads for Jerusalem confrontations)
By Dan Williams
JERUSALEM, July 4 (Reuters) - Palestinians infuriated at a kidnap and killing they blame on far-right Jews clashed with Israeli police in Jerusalem on Friday, while cross-border shelling in the Gaza Strip abated under Egyptian mediation.
"With our souls and our blood we will redeem you, martyr," chanted hundreds of mourners who gathered for the funeral of 16-year-old Mohammed Abu Khudair after prayers marking the Muslim fasting month of Ramadan.
His body was carried for burial through east Jerusalem wrapped in a Palestinian flag. Elsewhere in the city Palestinians threw rocks at police who fired back tear gas and stun grenades. There was no immediate word of casualties or arrests.
Three Israeli teenagers were abducted on June 12, triggering army sweeps of the occupied West Bank. Hostility boiled over this week with the discovery of their bodies and with Abu Khudair's slaying in what many Palestinians believe was revenge.
Short-range rocket salvoes from Gaza have drawn Israeli air strikes and a mobilisation of ground forces which Prime Minister Benjamin Netanyahu on Thursday threatened to use unless the enclave's dominant Hamas Islamists held fire.
Israel said three rockets and two mortar bombs were fired across the border from Gaza on Friday. Its Iron Dome interceptor shot down one rocket and the other projectiles struck open ground. Israeli army artillery then shelled southern Gaza. No one was hurt on either side, Israeli authorities said.
Hamas, whose armed cadres Israel accuses of carrying out some of the rocket launches this week, said it was in contact with Egyptian mediators trying to prevent further flare-ups.
"Hamas stressed that it was not interested in a confrontation and the occupation (Israel) was responsible for the escalation against our people and they have to stop it," Hamas spokesman Sami Abu Zuhri said.
Ahead of Abu Khudair's funeral, Israeli police deployed extra forces in Jerusalem and barred men under the age of 50 from al Aqsa mosque complex, the city's most sacred Muslim site.
Cairo, echoing Western powers, has also urged restraint by Israel over the killing of the three Jewish seminary students, which Netanyahu blames on Hamas men in the West Bank who are still at large. Hamas has neither confirmed nor denied involvement.
Israelis were incensed at the deaths and scores of ultra-nationalists went on anti-Arab rampages, scuffling with police. Hours after the three teens' televised funerals on Tuesday, witnesses said Abu Khudair was bundled into a car in East Jerusalem. His charred body was found in a forest on Wednesday.
Abu Khudair's family and other Palestinians, including President Mahmoud Abbas, said he was the victim of a Jewish vendetta and that Netanyahu's right-wing government bore responsibility.
Netanyahu called the killing "loathsome" and ordered a swift police investigation to find the culprits. Israeli authorities said they did not yet know whether Abu Khudair was indeed the victim of a hate crime.
The Israeli military said it had jailed four soldiers for posting anti-Arab messages on social media. A police spokesman said the force's cyber-crime unit was also cracking down on racial incitement online, whether by Jewish or Arab citizens.
While vowing to hit Hamas over the three Israeli teenagers' killings, Netanyahu is reluctant to launch a major operation in Gaza that could damage his already difficult ties with Abbas. Their U.S.-sponsored peace talks collapsed in April when the Palestinian leader agreed a surprise power-share with Hamas.
Yuval Steinitz, a cabinet minister close to Netanyahu, said escalation risked disrupting international efforts to negotiate a curb on Iran's disputed nuclear programme ahead of a July 20 deadline.
"We don't want, possibly, to divert all of the world's attention to something else now, of all times, from the matter of the Iranian nuclear programme, which is the existential threat to Israel - more than terrorism, more than riots," Steinitz told Israel's Army Radio on Friday. (Additional reporting by Nidal al-Mughrabi; editing by Andrew Roche)