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Which side receives precedence by a lefty, his left or right?
Some Poskim  rule that a left handed person is to give precedence to his right side, just like a right handed person. However others rule, based on Admur, that a lefty always gives precedence to his left side.
How should a lefty put on and tie his shoes?
Which shoe to tie first: Regarding which shoe one is to tie first, there is a dispute in whether a lefty is defined as a person who is left footed or a person who is left handed. The following is the ruling: If the person is left handed and also left footed then he is to immediately tie the right shoe, and then put on the left shoe and tie it. If he is left handed but not left footed then some Poskim rule he is to follow the same order as a righty in all matters, and he is thus to tie the left shoe and then tie the right shoe. Others however rule that he is to first tie his right shoe and then put on his left shoe and tie it. If he is left footed but not left handed the same dispute applies. Practically one is to choose one opinion to follow consistently.
 Beir Moshe 2/1; Piskeiy Teshuvos 2/4; Shaareiy Teshuvah 2/2 in name of Bechor Shor page 111; M”B 2/6 regarding wearing the right shoe first; see also chapter 4 Halacha 7B regarding washing hands and see there Peri Megadim 158 A”A 1; M”B 4/22 in name of Artzos Hachaim; Amudei Hashulchan 2/2; Kaf Hachaim 4/35; Miasef Lekol Hamachanos 4/53
 As based on Kabala the attribute of Chesed and Gevurah remains by the right and left of all people, even if one is right handed. [ibid; See Seder Hayom based on Zohar]
 Sefer Dinei Iter p. 43-46 based on Tzemach Tzedek Orach Chaim 4/6 and 5/8 regarding Netilas Yadayim, and based on Admur 183/7 that a lefty is to hold the Kos Shel Bracha in his left hand; 651/14 that a lefty is to hold the Lulav in his left hand; So writes also Rav Ginzberg in Haaros Ubiurim 783 p. 74
 See Shaareiy Teshuvah 2/6; M”B 2/6; Piskeiy Teshuvos 2/6; Halichos Shlomo 2/20
 Shaareiy Teshuvah 2/2 in name of Bechor Shor page 111
 Dinei Iter 2/6
 This is defined as whichever foot is stronger and hence used more when running, kicking etc. [Piskeiy Teshuvos 2 footnote 41]
 Shaareiy Teshuvah 2/2 in name of Bechor Shor page 111; M”B 2/6; As since the left [weak] side of a lefty is his right hand and on that hand he puts on Tefillin, therefore he should tie his right shoe first.
 Bechor Shur ibid; and so concludes Toras Chaim Sofer 3; Minchas Yitzchak 10/1
 M”B 2/6; Some versions of Shaareiy Teshuvah ibid support this understanding; So concludes Dinei Iter p. 61-65; See however Toras Chaim Sofer 3; Minchas Yitzchak 10/1
 Bechor Shur ibid rules that even in this case that he is right handed and thus places his Tefillin on his left arm, he is to nevertheless tie his right shoe first as in this regard the right is considered his left and the Torah preceded the left to the right regarding tying. However according to M”B 2/6 which depends the matter on the arm that the Tefillin are tied on, certainly he is to first tie the shoe on his left foot.
 Halichos Shlomo 2/20