Why do we celebrate lag baomer with bon-fires and bow-and-arrow. Why do we celebrate lag baomer at all?
From Halachically Speaking Volume 3 Issue 16, see there for sources, as well as other customs associated with Lag B'omer:
What Happened on Lag B’omer
Many different happenings took place on Lag B’omer. Some say it is the day Rav Shimon Bar Yochai came out of the cave that he and his son were in for thirteen years. Rav Shimon Bar Yochai died on Lag B’omer and revealed to us the Zohar. The Rama also died on Lag B’omer and many have the custom to go to his kever on Lag B’omer.
The Simcha of Lag B’omer
On Lag B’omer we are happy. Some say the reason is because the talmidim of Rav Akiva who died throughout the days following Pesach stopped dying on Lag B’omer. Some poskim ask if so many talmidim died why is his a reason to be happy? The answer may be we are happy that the talmidim which Rav Akiva acquired afterwards did not die. ....
Some say the mon started to fall on Lag B’omer. Others say the reason for the joy is based on reasons of kabbalah.
Some say the reason for bonfires on Lag B’omer is because when Rav Shimon Bar Yochai revealed the secrets of the Torah to us, he brought light to the world, so we make light in his honor. Others say the reason is as a remembrance of the fire that surrounded Rav Shimon Bar Yochai while he was in the cave.
Bows and Arrows
On Lag B’omer the custom of many is to shoot bows and arrows. One reason is because Hashem shows a rainbow (keshes) in the sky when He wants to destroy us, but the rainbow is the simon that he will not do so. In the generation of a few people a rainbow was not shown because of their merit. One of those people was Rav Shimon Bar Yochai. .... Some say since Lag B’omer is a day that is fit for one’s tefillas to be answered we shoot a bow and arrow. The beginning words of shema kol tefilaseinu spells keshes (rainbow).
Rema Mipano writes the reason for lag b'omer is that there are 24 days including lag b'omer where one doesnt say tachanun in the omer. Each day 1000 talmidim died. On the 24th day lag b'omer R Akiva was also supposed to die. But because of him not dying the gezairo stopped and no more died. That is why we keep lag bomer.
Without much in terms of early sources, many practices of some communities' festivities today may have origins that are not Jewish at all. For example, regarding bonfires, from Wikipedia:
In the Czech Republic, the festival called "Burning the Witches" (also Philp & Jacob Night, Walpurgis Night or Beltane) takes place on the night between 30 April and 1 May. This is a very old and still observed folk custom and special holiday. On that night, people gather together, light bonfires, and celebrate the coming of spring. In many places people erect maypoles.
The night between 30 April and 1 May was considered magical. The festival was probably originally celebrated when the moon was full closest to the day exactly between the spring equinox and summer solstice. People believed that on this night witches fly on the Sabbath, and indeed this is one of the biggest pagan holidays. People also believed, for example, in the opening of various caves treasures were hidden. The main purpose of this old folk custom was probably a celebration of fertility.
To protect themselves against witches, people lit bonfires in high places, calling these fires "Burning the Witches". Some people took to jumping over the fire in order to ensure youth and fertility. The ash from these fires supposedly had a special power to raise crops, and people also walked the cattle through the ashes to ensure fertility.
Likewise, it seems conceivable that customary archery events may have origins in the competitive events and spectacles, e.g., of the Roman pagan festival of Floralia.
(See also Mitchell First hattip: DoubleAA)