Doing Kaparos on another person:

By Kaparos, must each person hold the chicken and wave it over his/her own head or may he/she have someone else hold it and wave it over his/her head?

It is initially preferable for the chicken to be held and waved by the person on whose behalf the Kaparah is being performed.[1] Nonetheless, in a time of need, such as children, or the queasy minded who cannot hold a chicken, another person may hold it.[2] [A husband may perform the circling over his wife who is a Niddah so long as they are careful to avoid any contact.[3]]

Who is to say Zeh Chalifasi when doing on behalf of another? The person who the Kaparah is being done for is to say “Zeh Chalifasi” while the circles are being made over their head.[4] Alternatively, the person who is holding and circling the chicken is to say “Zeh Chalifascha”.[5] This Nussach is to be said when doing Kaparos on behalf of children who are too young to recite the Nussach. 

How is Kaparaos to be performed to small children who cannot do so themselves?[6]

The father is to do so for them, as explained above.

Must one first perform Kaparos on himself prior to performing it on others [i.e. encircling the chicken over the heads of others]?[7]

It is proper for one to first perform Kaparos on himself prior to doing so on behalf of others. If there is a reason that he is not able to do so, then one may initially do the Kaparos for others before doing so on himself.

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[1] Simple understanding of Admur 605:1; Siddur; See Kitzur SHU”A 131:1 “Each one is to take their own Kaparah in their right hand”; Mateh Efraim 605:6 “If one is doing the wave for someone else because they don’t know”; Makor Chaim 605; Kitzur Shelah; Siddur Yaavetz

Other opinions: Some Rishonim record that the Minhag is for the Rav of the town to hold the chicken and wave it around each person’s head. [See Likkutei Maharich]

[2] Kitzur SHU”A 131:1; Mateh Efraim 605:6

[3] See Nitei Gavriel 11:10

[4] See Nitei Gavriel 11 footnote 17

[5] Kitzur SHU”A ibid

[6] Likkutei Maharich

[7] Mateh Efraim 604:6