In the first incident, witnesses said Israeli soldiers backed by helicopters entered the northern Gaza town of Bait Hanun and engaged in heavy clashes with Palestinian fighters there.
Residents said the operation targeted the house of a member of Hamas. A Palestinian father and his son were killed and at least four others were wounded, medics said.
The army said two fighters opened fire on Israeli forces from within the building, prompting soldiers to return fire.
An Israeli army spokesman added that two Hamas fighters wanted for firing rockets into Israel and setting off explosives were also arrested by Israeli special forces from the house.
The house was later demolished by an army bulldozer, residents said.
At the same time, Israeli air strikes in a nearby part of Bait Hanun wounded three Palestinians, two of them children, with shrapnel, medics said.
In the second incident, the Israeli army said it had shot a man south of the Kisufim border crossing in central Gaza overnight. They said the man looked like he intended to infiltrate Israel. Palestinian medics recovered the body.
Gaza is in the grip of lawlessness
Later on Saturday, two Palestinians were lightly injured by flying debris from an air strike on a building in northern Gaza, residents said.
An Israeli military spokesman said a weapons-storage facility was targeted.
The attacks were the latest in an eight-week Israeli offensive in Gaza that started after fighters abducted an Israeli soldier in a raid.
More than 200 Palestinians, about half of them civilians, have been killed during the offensive.
As well as seeking to recover the captured soldier, the operation is aimed at preventing fighters from firing makeshift rockets into Israel from northern Gaza. Rocket fire has resumed sporadically in recent days.
In a sign of continuing lawlessness in the narrow coastal strip, a Hamas security officer was killed in clashes with a local clan in the southern town of Khan Yunus, medics and witnesses said.
Five members of the clan were also injured.
Residents said security forces had been trying to defuse a feud between rival clans in the area.