Removing cobwebs before Shabbos:
It is proper for one to clear his house of [spiders and] cobwebs [i.e. Korei Akavish] before Shabbos, in order so one’s house is clean in honor of Shabbos. [Preferably, this should be done prior to the 5th hour of the day on Erev Shabbos.]
The Kabalistic reason for cleaning the cobwebs:
The author of the Reishis Chochma states that the cleaning of the cobwebs from the house on Erev Shabbos contains a Kabalistic secret. The Shlah Hakadosh explains this as follows: One’s home is considered similar to the Sefira of Malchus which is surrounded by thorns, which are the Reshaim and evil Kelipos. In the home, these Kelipos are manifested as the spider webs and they search to cause discord within the family and destroy the Shalom Bayis. One is thus to remove the spiderwebs from his home, in order to remove the idolatry from the land. This especially applies prior to Shabbos, as Shabbos is called Malchus.
Is one to kill the spider?
Although the Arizal was careful never to kill any living creature, and so was likewise the custom of the Rebbe Rashab, some Gedolei Yisrael were particular to make an exception to spiders and would actively kill them due to their representation of Kelipa.
May one remove cobwebs on Shabbos?
 Admur 262:2; M”A 262; Olas Shabbos 250; Shelah Emek Habracha Miseches Shabbos 130b; Totzos Chaim [Kitzur of Reishis Chochma, written by the same author] Letter Tzadik, Hanhagos Baal Teshuvah 11; Or Tzadikim 28; M”B 250:3 and 262:3; Kaf Hachaim 250:10; Divrei Torah [Munkatch] 9:99; Piskeiy Teshuvos 250:9
 Lit. good
 Shlah Emek Habracha Shabbos 130b
 Kaf Hachaim ibid based on Shelah ibid
The reason: As the spider web represents the Kleipos, and starting from the 5th hour of the day on Erev Shabbos shines the ray of Shabbos [Shaar Hakavanos p. 62a, and hence the Kelipos should be removed prior to this time. [Kaf Hachaim ibid]
 Kaf Hachaim ibid; See also Darkei Chaim Veshalom 1080
 In Totzos Chaim ibid
 In Emek Habracha Miseches Shabbos 130b
 Shaar Hamitzvos Noach; Shaar Hagilgulim 38; Torah Leshma 397; See also Kav Hayashar 83 “One should not kill any creature for no reason, even if it can cause one pain, so long as it is not chasing after you”
 Reshimos 154:473
 Darkei Chaim Veshalom 1080