In the first incident, a Palestinian rammed a group of Israelis with a car at a junction, wounding four of them, and was then killed by security forces, police said.
The alleged attacker was said to have driven the car into an area south of Nablus where Israelis typically gather to hitch rides. A checkpoint is located nearby and Israeli border police opened fire.
Israeli public radio described two of the wounded as in a serious condition.
The incident occurred at the entrance to the Beitar Illit settlement south of Jerusalem. The fate of the woman was not immediately clear.
Also Read
A wave of knife, gun and car attacks have hit Israel and the Palestinian territories since the start of October.
The violence has left 73 Palestinians dead, around half of them alleged attackers. Nine Israelis have also been killed as well as one Arab Israeli.
Violent protests have also erupted in annexed east Jerusalem, the occupied West Bank and the Gaza Strip.
The latest violence came as Israeli Prime Minister Benjamin Netanyahu prepared to leave for Washington on Sunday to meet US President Barack Obama.
The meeting will be centred on US defence aid to Israel following the July nuclear accord between major powers and Iran, but the recent Israeli-Palestinian violence is also expected to be discussed.
At the start of a cabinet meeting on Sunday, Netanyahu said he will speak with Obama about "possible progress with the Palestinians, or at least stabilising the situation with them, and, of course, strengthening the security of the state of Israel".