I don't know of specific passages in the Talmud that talk about Zionism; however, I do know that it doesn't outright contradict it. The more important book in Judaism is the Tanakh (basically the "Old Testament" for you non-Jews) and in there Zionism is very much supported. So, what I can say without doubt is that the Talmud is in favor for a Jewish State in Israel. What is up for debate is the conditions for which that State may arise, which is where you get disagreement from some religious Jews.
Essentially, the basic idea of Zionism - that Jews deserve a State in Israel - is supported but not everyone agrees with the conditions for reestablishing the Israeli nation. For example, Neturei Karta is often held up as "anti-Zionist" but they oppose Israel primarily because they require obvious divine intervention in reestablishing that nation, at the moment of the new iteration's creation. In essence, they're not totally against Zionism, but against the popular interpretation of it today. When it comes to the Talmud and Zionism, it may agree with a Jewish State but as different people have different interpretations we see that the Jews aren't all unified. Trust me, that's nothing new throughout the many topics in the Talmud or throughout anything in the Jewish world. Jews have literally been arguing about the same texts for thousands of years.
Note that the Zionist movement isn't a monolith, and different Zionist Jews have different opinions. Many Jews aren't as supportive of the settlements as you may think yet will be unshaken when it comes to green-line Israel. Most Jews are between the settlers and the anti-Zionists and they all have different interpretations of the Talmud, though if you force that binary decision then I'll say that most interpret the Talmud much more in favor of Israeli settlers. Which, if you are one of those who labels all Israelis as "settlers" then clearly much more will answer "supports settlers" to your second question. Ultra-Orthodox Jews known for condemning Israel (since many pro-Israel ultra-Orthodox Jews will still condemn Israel over many issues) are a MINORITY of Jews anyway, so this isn't surprising.
Edit: The thing with Korea is that they viewed Talmudic study as a way to improve cognitive processes and mental strength. At least, that's how I understood it. It really amounted to a fad for them and it's not too common for a Korea to even know what a Talmud is. Their viewpoint of the Jews is very favorable, and they admire the Jewish peoples' reputation as doctors, lawyers, leaders and bankers. They attributed that success to Talmudic study, so that's why some picked it up.