Avram:
As to your other questions:
To answer your second question, let’s determine the goal of the prayers.
If your goal is to worship the idolatry of Zionism then praying for the State makes sense, as, unfortunately, our MO brethren have gone so far as to insert into their siddur.
If, however, your goal is to beseech G-d to protect the Jews who live in E”Y simply because they are our Jewish brethren (especially if you also understand that the State of Israel is a tremendous affront to Him), then the most logical conclusion is to simply pray for the safety of our brethren in E”Y and leave the “how-to” to G-d.
Then, if Hashem chooses to do so through the IDF, then that’s His choice. If, instead, He does so by bringing Mashiach now and ending this nonsense and kefira immediately and bloodlessly, that would, of course, be much better.
This was the same mistake made at the time of Israel’s founding. The Brisker Rav said that some observant Jews prayed for a State as their salvation in 1948 instead of for Hashem’s true salvation, and Hashem answered their prayers in kind, with the utter disaster we have seen, Hashem yiracheim.
So it would be very wise to not play G-d and simply pray for the well-being of our brethren and leave to G-d the running of His world.
Regarding politics, there is no question that Israel is held to a different standard than the standard to which the goyim hold (other) goy nations. Notice that all goyim refer to Israel as the “Jewish State” when Israel is, of course, a Zionist state and therefore not a Jewish State despite some Jewish trappings (somewhat like a chazir trying to vainly show it is kosher).
It seems pretty clear that Zionism is the reason for this treatment by the nations. Bilam stated (and Hashem put his words into the Torah) “Hein Am Livadad Yishkon uVaGoyim lo yischashav”. The Zionists are, of course, kofrim in this, too, while Hashem allows the nations of the world to make the Zionists live those words in constant and living refutation of their kefirah.
Regarding working for the State of Israel, while seemingly very ill-advised for any number of reasons, this is clearly an halachic issue that requires the advisement of an LOR.