Is The Jewish Home (political party) authentically Dati Leumi? Do they actually make progress in the Knesset? Is it permissible to believe in the Greater Israel and expect to gain back our land? Why or why not? What is the likely hood of their candidate becoming PM in the near future? Is it two separate parties (ie why do they appear to have two main candidates according to the wikipedia page)?

Yes, what's now called: "Yemina", is the most dati-leumi party around, even though Ayelet Shaked herself, the head of the party isn't personally Shabbat-observant. This is because for political reasons (so as not to waste votes for small parties which may not pass the necessary minimum 4 seats to get into Knesset), three parties united (Bayit Yehudi- headed by Rabbi Rafi Peretz, HaIchud HaLeumi headed by Betzalel Smutrich, and HaYemin HaChadash- headed by Shaked and Naftali Bennet). In my previous answer, I related to how the God of History will eventually return us the Greater Israel. The process of Israel becoming more religious and nationalistic, is a slow, natural democratic process and cannot be any other way, as I wrote you previously. I can't see the Yemina candidate being PM in the near future, but again, nobody foresaw the sudden collapse of the Soviet Union (bringing us more than 1,200,000 olim), or the election of someone like Trump, or the 6 Day War, and often God suddenly hastens the process, so you never know when...