Okay, so I went to research this out carefully. The off the cuff answer is yes, we should support Israel, or else we will suffer the consequences:
This is a promise to Abraham concerning the nation of Israel.
Also, we are called, as Christians, to respect the governments that have been placed over us. So if I went to Isreal, I would be responsible for obeying the Isreali government, as per Romans 13. So being "pro-Palestinian" and going over there to incite violence against the Isreali government would not be biblically condoned for Christians. To the extent that the Palestinians want to rebel against the Isreali Government, supporting them is not okay.
However, I don't have to agree with every decision made by the Isreali government, especially since I don't live there. For example, I have heard about people in Isreal who lived there when the government took control who have been forced out of their homes and made to move away.
This is a violation of God's law, notably that the Isrealites were not supposed to oppress aliens who were living among them, because they were slaves in the land of Egypt.
So if the Jewish people wish to live in the land of Isreal, they should treat the people that were there before them with human decency. Unless they are rebelling against the Government, in which case the Isreali government is well within its rights to defend itself.
The Two-state solution was proposed by the British, who are great at solving conflict. (NOT) See the Balfour Declaration information here: [https://www.history.com/topics/middle-east/history-of-israel] The British are also responsible for the fun conflict known as Iraq. Note that the Palestinian Liberation Organization (PLO) supports the two-state solution. This is their perspective on the whole thing here [https://www.aljazeera.com/program/episode/2009/8/12/timeline-history-of-a-revolution/] [https://www.history.com/topics/middle-east/palestine] However, at this point (and this is just my perspective) the two-state solution really seems to be a false hope. I (personally) would support a one-state solution in which the Palestinians are allowed to return to their homes if they submit to the Isreali government and agree not to kill any more people or blow any more things up.
(The Arabs, however, seem to be very into killing people and blowing things up in general, in case you haven't seen the news, so the Isreali government probably isn't inclined to believe that they will ever submit to the government. This is the Isreali government's perspective: [https://www.mfa.gov.il/MFA/AboutIsrael/History/Pages/HISTORY-%20The%20State%20of%20Israel.aspx])
This would get people out of the refugee camps. If you want to know what it's like to live in a refugee camp, check out the book When Stars Are Scattered. That is a graphic novel with comics and pictures. You'll probably like that one. People can spend their entire lives in refugee camps - it's not fun. Another book to understand what is going on in Palestine is called Outside The Ark, which talks directly about that experience.
So it's technically possible to be "on neither side" as a Christian. In fact, as Christians, we technically are on neither side by default. I also would see no occasion to remark on those who feed and clothe Palestinian refugees or treat them like human beings or even supporting a return of these people to their homes. By all means, I would see them accept the Gospel as much as anyone else. But, as I said before, there is no place for us supporting a campaign of violence against the Isreali government. That isn't biblical.
For this one, I defer to the Jewish scholar Dr. Arnold Fruchtenbaum to explain. Your knowledge is correct - the Antichrist will be making a peace treaty with Isreal, and the current state of Isreal is a regathering for God's judgement.
He then quotes Ezekiel 20:33-38 to explain the regathering in unbelief.
He also quotes Ezekiel 22:17-22:
In addition, Christians are not called to support the Antichrist and are to defy him. But does all of this mean that Christians should withdraw support from the Jewish state? Do you support God's judgment? Um...I think that is the wrong question.
God does lot of things without the assistance of Christians - we're not the only agent of his sovereignty in the world. Regardless of whether we support this state or not, it is, because God has ordained it. Ordained it for judgement, yes, but he has ordained it. We are not called to oppose God's sovereignty. Aside from our call to proclaim the Gospel, which we should do for the Isreali and Palestinian alike, it is not something Christians need be concerned with. It simply is.
It is an interesting piece on the eschatological roadmap, and since you find eschatology fascinating, I would recommend Footsteps of the Messiah. Perhaps you can find an audiobook version of it for your walks. Parts of it are actually funny, including the part of it that I based my defunct Christward RPG Babylon Invaders on.
Which is a good segway to my next post, which may well be about the two witnesses and Greek grammar. Lovely. But that's enough for now.
(Also if anyone knows more than what I dug up from my research about the Jewish state in Isreal I'd be happy to hear about it. This is just what I could find/have to answer the questions.)