What exactly is the story with the "lishma" requirement regarding machine-made tzitzit strings? Is it enough that they are manufactured specifically for the purpose of being used for the mitzvah, even if it’s not the Jew himself operating the machine? Or does the Jew himself, who has in mind that it should be used for performing the mitzvah, need to be the one operating the wool-twisting machine? Thanks in advance!

Tzitzit strings must be spun for the purpose (lishma) of the mitzvah of tzitzit. There is a general arguement between the Rambam nd Rabbeinu Asher if a non Jew performs a task instructed by a Jew to do it "lishma" and obsers him, if this is sufficient. We generally do not rely on the lenient position, since we are dealing with a Torah rule (the requirement of "lishma") As to the question as to whether there is a possibility of "lishma" in a machine process at all, this applies to matzot on Pesach, which must be baked for the purpose of eating Matzot on Pesach, or parchment for tefillin or a sefer Torah which must be tanned for the holiness of those purposes, the custom is to allow relying on the lenient positions that machine "lishma" is considered "lishma" but it is not considered a preferred method if hand made alternatives exist. Price is an issue, and certainly one is allowed to follow the lenient position if it would be difficult to buy tzitzit, but I don't think the price differential is that great for most people.