Why does Tim O'Brien (as the writer) "kill" Ted Lavender and not any other character?What is significant about his death?

I'm not sure what you mean by "kill."  O'Brien doesn't use
the active verb "kill."  It's the passive verb phrase "was shot," "was dead," and "was
shot and killed."  All of it is in passive voice.

The first
mention of Lavender's death is on page 2:

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Ted Lavender, who was scared, carried
tranquilizers until he was shot in the head outside the
village of Than Khe in
mid-April.

The death is
described matter-of-factly, in passive voice.  O'Brien describes the death as if it were
an everyday occurrence, like the rain or a new mission.  It's part of war.  It's
something else to carry.

There is no one physically
responsible for the death in Alpha Company.  We must assume that a VC sniper took
Lavender out with a head shot.  Morally, all the men feel guilty for it, Cross the
most.

Later, the third-person omniscient narration says on
page 7:

...now
Ted Lavender was dead because he lover her so much and
could not stop thinking about
her.

So, it is Lt. Jimmy
Cross who feels the most guilt for Lavender's death, not O'Brien.  He carries the weight
of the dead body literally and figuratively (on his
conscience):

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But Ted Lavender, who was
scared, carried 34 rounds when he was shot and
killed outside Than Khe, and he went down under an exceptional burden,
more than 20 pounds of ammunition, plus the flak jacket and helmet and rations and water
and toilet paper and tranquilizers and all the rest, plus the unweighted
fear.

O'Brien has Lavender
killed to serve several purposes: Lavender is a symbol of unweighed fear.  The function
of Lavender is to be a doppelganger, a ghostly twin to haunt Cross (a Christ-figure),
the one who feels the most guilt for his death.  The novel begins with a death here,
much like Hamlet does with the Ghost, and Lavender's death hangs
over the story and novel as a whole.  Like the Ghost in Hamlet,
Lavender will keep reappearing.  A major motif, as you know, is the ghost in
The Things They Carried.  Observe some of the titles of the other
stories: "Ghost Soldiers"; "Lives of the Dead"; "The Man I
Killed."

The purpose of storytelling, according to O'Brien,
is to bring the dead back to life through memory: it is to resurrect ghosts.  That is
both the painful and joyous task of a writer: to turn a war story into a love story by
honoring the dead.