jump x lines after match

replace next line after line match

sed '/^3/{ n; s/.*/x/ }' <(seq 7)

replace second line after line match

sed '/^3/{ n; n; s/.*/x/ }' <(seq 7)

append/insert a new line after match

sed '/^3/a new_line' <(seq 7)

jump 1 line after match and append a new line

sed '/^3/!{ p; d }; n; a new_line' <(seq 7)

jump 2 lines after match and insert a new line

sed '/^3/!{ p; d }; n; n; a new_line' <(seq 7)