Must one wash both hands if only one hand touched the shoe?
One is only required to wash the hand that touched the shoe, although one who is stringent to wash both hands is blessed. According to all one must wash the entire hand that touched the shoe.
Must one wash his hands if he removed his shoe without using his hands?
Must one wash his hands if he removed his shoe using a cloth?
Must one wash his hands after touching a new shoe that has never been worn?
No. Thus one is not required to wash his hands after touching new shoes in a store.
Must one wash his hands after touching the top of a boot?
Some write one does not need to wash his hands being that the top of the boot does not become dirty and thus does not have an evil spirit. However practically this matter requires further analysis.
Must one wash his hands if he touched his shoelaces?
Some Poskim are lenient to not require washing the hands if one merely touched the shoelaces.
Must one wash his hands after touching his socks?
No. However there are those which are stringent in this matter. If the socks are filled with sweat then one is to wash his hands according to all.
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 128/27; Tehila Ledavid 4/18; Poskim in Kaf Hachaim 4/71; Aruch Hashulchan 4/21; It makes no difference whether the shoes are his or not or whether he is currently wearing them or not. [Piskeiy Teshuvos 4/22]
 Kama 4/18
 128/27; Poskim mentioned in Kaf Hachaim 4/70; being an evil spirit resides on oneself upon touching shoes and hence the hands need to be washed due to danger. [Admur ibid]
Other Opinions: Some Poskim rule that touching the shoes do not bring an impure spirit and hence do not require washing with water. Rather the hand requires cleaning in any fashion in order to be allowed to learn Torah. [Machatzis Hashekel 4/17; Nivei Shalom 4/12; Machazik Bracha brought in Shaareiy Teshuvah 4/12; Soles Belula 4/14 in name of Makor Chaim; Ben Ish Chaiy Toldos 17] Admur however rules that it causes impurity, and hence only washing with water helps. So also rules Kisei Eliyahu ibid; Yifei Laleiv 4/18 in name of Mor Uketzia; Peri Megadim 4 M”Z 14; and so is implied from many Poskim. [brought in Kaf Hachaim 4/60]
 Ketzos Hashulchan 2/11; Kaf Hachaim 4/64; A”A 4 M”Z 18
 Kaf Hachaim 4/86 regarding uncovered areas [see F Q&A]; Piskeiy Teshuvos 4/22 and footnote 179
 Admur 4/18 “hand”; M”A 4/19; Shiyurei Kneses Hagedola 128/4; Elya Raba 4/13; Mamar Mordechai 4/9; Chayeh Adam 2/5; Kaf Hachaim 4/69; Nimukeiy Orach Chaim 4/3
Other Opinions: Olas Tamid 4/12 [brought in Peri Megadim 4 A”A 19; Ashel Avraham Butchach] rules one is required to wash his hands after removing the shoes even if he did not use his hands. We do not rule like this opinion.
 Yifei Laleiv 1/27 brought in Kaf Hachaim 4/87; See Admur 97/3; However see Olas Tamid in previous footnote.
 Ben Ish Chaiy Toldos 17; Kaf Hachaim 4/71; Mor Uketzia; Daas Torah; Shearim Hametzuyanim 2/9; Poskim mentioned in Piskeiy Teshuvos 4 footnote 214
 Piskeiy Teshuvos 4/22 footnote 215
 Nimukeiy Orach Chaim 4/3
 Ashel Avraham Butchach 4; Nimukeiy Orach Chaim 4/3 negates this opinion. See also Aruch Hashulchan 4/21; Piskeiy Teshuvos 4/22
 Piskeiy Teshuvos 4/22 in name of Chazon Ish; Halichos Shlomo 20/18; Yabia Omer 5/1
 Kaf Hachaim 4/72 in name of Poskim; Ashel Avraham Butchach; Aruch Hashulchan 4/21
 Brought in Kaf Hachaim ibid
 Sheivet Halevy 8/2; See Halacha F Q&A