Israel says its jets hit two targets in the Gaza Strip after rockets were fired into southern Israel.
Israel's military said it targeted Hamas weapon-making facilities. Palestinian officials said a woman and child in a nearby house were killed.
West Bank officials say a boy of 13 was shot dead by Israeli forces during clashes near a Jewish settlement.
And later four Israelis were injured in the latest stabbing attack as tensions continued to escalate.
Earlier in the West Bank, a Palestinian woman was seriously injured after setting off an explosion at a checkpoint, Israeli police said.
At least 24 Palestinians, including attackers, and four Israelis have been killed in violence so far this month, news agencies report.
This follows weeks of tension over access to a site in East Jerusalem sacred to both Jews and Muslims.
Palestinians fear Israel plans to change arrangements at the al-Aqsa mosque/Temple Mount compound, where Jews are allowed to visit but not allowed to pray - something Israel insists it will continue.
The Israeli military "holds Hamas responsible for any act of aggression from the Gaza Strip." said army spokesman Lt Col Peter Lerner following Sunday's air strikes.
The air strike destroyed a house in Zeitun, northern Gaza, killing a 30-year-old pregnant woman and her daughter aged two.
A spokesman for Hamas, which controls Gaza, said "this shows the occupation's desire to escalate", AFP reported.
"We warn the occupation against continuing this foolishness," it quoted Sami Abu Zuhri as saying.
In the West Bank, the Israeli army said the teenager was killed near Ramallah as security forces clashed with Palestinians trying to get to a road to attack settlers' cars.
Dozens of Palestinians were reported to have been shot and injured in a number of confrontations in the West Bank.
Later in the day, an Israeli Arab drove his car at a group of people near the northern Israeli town of Hadera, then stabbed four of them, police said.
One, a 19-year-old female soldier, is said to be seriously injured.
Early on Sunday, Israeli police believe they foiled an attack after stopping a woman driver near another West Bank settlement, Maale Adumim.
As she walked toward police officers, there was an explosion in the vehicle.
The Shin Bet internal security agency said she tried to ignite a gas cylinder using flammable materials.
"We are not talking about an explosive device," a statement said.
Late on Saturday, US Secretary of State John Kerry expressed "deep concern" over the situation.
He made separate phone calls to Israeli PM Benjamin Netanyahu and Palestinian Authority President Mahmoud Abbas.
Mr Kerry "stressed the importance of upholding the status quo in word and deed" at the site, the state department said.
He offered his support in efforts to restore calm, but both men blamed the other side for the rising violence.
The violence has spurred talk from Hamas, which dominates Gaza, of a new Palestinian intifada, or uprising.
But the clashes have not yet reached the scale of previous intifadas, with no clear mass movement or leadership so far emerging.