Suspected separatist fighters and the soldier were killed in two unrelated skirmishes in the southern Kashmir district of Poonch on Sunday evening, a police spokesman said from the summer capital.
"The fighting broke out when troops ringed two villages on a tip-off that militants were present there," the spokesman said.
A civilian was killed and two others were wounded on Monday when Border Security Force (BSF) personnel opened fire on a car after it had ignored a signal to stop in the northern town of Magam, a BSF spokesman said.
The BSF said it fired on the car after being shot at first. A separatist leader in Srinagar, Muhammad Yasin Malik, disputed the BSF version.
Kashmiri separatists had called
for a boycott of the civic polls
The violence came after a third round of civic elections ended peacefully on Sunday in the two sensitive districts of Anantnag and Pulwama.
Separatist politicians and rebel groups had called on voters to shun the elections, the first in more than a quarter century, saying the polls were no substitute for the right to self-determination.
Tens of thousands of people have died in Kashmir since the eruption of civil conflict in 1989, with Kashmiri separatist groups arguing for independence from Indian rule.