- Those that have added the saying of Hallel but without the recitation of a bracha (benediction)
- Shuls that have added Hallel with the bracha
- Shuls that have not added anything.
R' Howard Jachter lays out and analyzes a lot of relevant sources on the Hallel question here. It's worth reading the whole essay; it packs a lot of material into a short piece. His conclusion is:
It is difficult to arrive at a definitive conclusion whether one should recite Hallel with a Beracha on Yom Haatzmaut. Hence, most of those who recite Hallel on Yom Haatzmaut omit the Beracha. The practice of reciting Hallel without a Beracha upon a miracle is mentioned in the Meiri’s commentary to Pesachim 117a. Moreover, the practice of reciting Hallel without a Beracha is familiar to Sephardic Jews who follow the opinion of the Rambam (Hilchot Chanukah 3:7) to recite Hallel without a Beracha on Rosh Chodesh and the last six days of Pesach. Far from being a “cop-out,” the approach of reciting Hallel without a Beracha is an expression of a sophisticated recognition of both the positive and negative aspects of Medinat Yisrael. We are full of joy that Medinat Yisrael exists, but we are pained by its shortcomings.
Finally, every few years I re-read and am brought to tears by "The Significance of Yom Ha'atzmaut" by R' Mordechai Greenberg, the head of Yeshivat Kerem B'Yavneh. (Sorry about being a day late with this; my old link for it was broken, and I had to track it down.) It starts off relatively slowly with a theory of uniquely Jewish nationalism, but by the time you get to the end, you'll want to run to Israel right now and be part of the Redemption. At least, that's my reaction.
It depends on how you define a "chag,"re. an issuer melacha. It's worth noting that the greeting on Yom Ha'atzmaut is "Moadim LeSimcha liGeulah Shleimah I never quite understood this, but in light of the definition of "chag" as a day on which Melacha is prohibited, perhaps this is the origin of the usage of the phrase Moadim LeSimcha (the greeting for Chol HaMoed).
Depends on how you are using that word. It's an Israeli holiday and in Hebrew the word for holiday is chag. If one were speaking about Thanksgiving in Hebrew you would also call it a chag.