Can one follow the lenient opinions of the Aruch Hashulchun and the Chasam Sofer regarding whether a gentile is forbidden or allowed to spay/neuter animals? and the resulting idea of neutering a male pet as long as the procedure is performed by a gentile?
The Shulchan Aruch writes that it is forbidden to neuter an animal even by means of requesting a non-Jew to do so (Even Ha-Ezer 5:14).
This is either because the prohibition applies even to a non-Jew (see Beis Shmuel who holds this to be true, whereas the Aruch Ha-Shulchan disputes this), or because of the prohibition of requesting a non-Jew to perform any prohibition.
However, for a female animal there is room for leniency where there are special circumstances. In addition, if done by indirect means (preventing blood flow from the reproductive organs) poskim are lenient in asking a non-Jew.
See Nishmas Avraham (Even Ha-Ezer 5:9), citing from Rav S. Z. Auerbach; Shevet Ha-Levi (Vol. 6, no. 204); Har Zvi (Yoreh De’ah 23); Shevet Mi-Yehuudah 4:18.