Police said 16 people were wounded in the clashes across the strife-torn Himalayan region late on Wednesday and on Thursday.
In one incident, two fighters tried to raid an army camp in Poonch district, south of the capital Srinagar, but were killed. A woman was also killed and four children were wounded.
None of the dozen or so Muslim separatist fighters groups claimed responsibility for the attack.
Elsewhere, 12 fighters, two civilians and a soldier were killed in separate clashes across the region, police said.
The fresh wave of violence in Kashmir is expected to slow tentative peace moves between nuclear-armed India and Pakistan.
“The situation seemed to us slightly improved”
But a European Union delegation on a five-day visit to the region said it believed the level of violence had fallen.
“The situation seemed to us slightly improved,” Benedetto Amari, head of the delegation, told a news conference in Srinagar.
The delegation had talks with state government officials and representatives of an alliance of separatist political groups.
India and Pakistan have fought three wars since 1947, two of them over Kashmir, and nearly went to war again following an attack on the Indian Parliament in December 2001 that India blamed on Pakistan-based fighters.
The South Asian neighbours recently restored full diplomatic relations and cross-border bus links in the first steps toward normalcy.