If Palestine got to have its own recognized (by everyone including Israel) and sovereign state, what other different current nation would it most likely resemble?

Palestine

Obviously similar to neighboring countries like Egypt, Lebanon, Syria, Jordan.

Hint: It is obvious you are trying to propagandize in favor of Israel. No, no possible answer to your question justifies anything Israel has been doing.

They would definitely have some kind of mandatory military service instituted. I don't think they would ever trust their Zionist neighbors.

Culturally and economically speaking, it would be very similar to Lebanon or Syria in my opinion.

It depends on how that country is formed, and what happens to Jerusalem.

If Israel withdraws from west bank and Gaza, and truly allows them to be a sovereign state(s), with their own navy, air force, ground force, then the US would step in via the UAE. I would also guess that they will get lots of money coming in from other Arab governments because they all want influence in Palestine, in addition to money from Arabs as people.

In terms of government, if elections are held I assume Hamas would win (in Gaza and wb). In which case countries like Egypt will start being less friendly towards Hamas than they are right now, because they can be seen as a viable government. The reason Egyptian government would dislike Hamas is because they are a spin-off of the brotherhood, which the sisi couped in order to gain power. Generally, Egyptian government is very afraid of the brotherhood and will root them out wherever possible because they are often the only party (other than police and military) that may be in a position to rule the country.

In terms of long term economy and how the government would function I have no idea. Hamas controls an area the size of a small town, and the Palestinian authority has no real autonomy, its not even allowed to collect taxes. My guess tho is that it will resemble south Africa in terms of infrastructure (garbage).

I don't think it will collapse like Libya and be run by criminals, but it won't be as prosperous and safe as many of the Gulf countries.

Basically I'm talking out of my ass, there's really no way to know

Depends on the starting conditions.

There are two main "forces" at play here:

Hamas, which is an fundamentalist, religious and military organization backed by Iran. If they were to gain power in a Palestinian state, it would look something like Hezbollah controlled areas in Lebanon. So... not good.

Fatah, on the other hand, is a (relatively) secular organization that's in good relations with western countries. If they were to gain power, Palestine would be more open to western influence, and will probably treat women, secular people and minorities better. This version of Palestine will probably be the most pro-western Arab state, so it might be more influenced by western values more than other Arab states. Of course, in the mid- or long run it's possible an extremist power will rise regardless of western backing (ex. Iran).

Assuming a you're talking about the near future, which organization will have control largely depends on if Hamas would exist. If so, they'll probably get the credit for a recognized state due to their "resistance". Then again, it's very possible one of the conditions for a universally recognized state will be the elimination of Hamas as a political (and obviously military) force.

I'm kinda ignoring the "including Israel" part of your question, as Israel would absolutely not accept any version of Palestine with Hamas.

I’m interested in how a free Palestine would play out in terms of what that would mean for women, gays, children, people of non-muslim religions, in terms of personal freedoms, etc.

I think this is a ludicrously speculative question given the current state of things, and possibly a leading one. It smells like imperial core feigned human rights concern for the purposes of manufacturing consent. That’s the reason we hear about Palestinian human rights from Western governments, human rights NGOs, and corporate media; and therefore why they’re salient in the public discourse. It’s propaganda. Citations Needed podcast: The Human Rights Concern Troll Industrial Complex

what that would mean for women, gays, children, people of non-muslim religions, in terms of personal freedoms, etc

So the wonderful thing about anti-colonial movements is that even if they start right wing they have to get more left wing as time progresses, simply due to the power dynamics that are created through the process of throwing out the imperialists. Once you stop foreign oppression, you have a grassroots mass of militant and armed revolutionaries that you're accountable to and who get upset when you try to reproduce the same economic system. You also are tied at the hip to them as the imperialists seek a return to an old system.

This includes movements where there is no significant communist presence, but it happens faster when there are communists.

Luckily, the PLFP is the second biggest party represented in the Palestinian joint operations room and have significant military and political power, meaning that in Palestine it probably will take a lot less time to materialize.

Japan

Probably a bit like a combo of Lebanon and Jordan, just my guess. Both have diverse populations and balance modernity with tradition. Of course, it all depends on how they decide to structure their society and government.

I’m interested in how a free Palestine would play out in terms of what that would mean for women, gays, children, people of non-muslim religions, in terms of personal freedoms, etc. For the average citizen/denizen what would that look like?

Well the future doens't permit accurate predictions.

Ahmad Yusuf, the deputy foreign minister of Hamas, said in 2010 that he wanted a country like Turkey, not like the Taliban:

According to Yusuf, who said he is writing a book called “Erdoğan and a New Strategic Vision,” the Taliban is “opposed to everything,” including education and women’s rights. “Erdoğan’s model, on the other hand, is liberal. It is a model that dares to take responsibility and change things and establishes good relations between the religious and secular elements of society,” he said. “It is a model that works for democracy and human rights, and supports an open society. That is what we want.”

inb4 Woids saying Erdoğan isn't liberal enough for them: homosexuality is not criminalised in Turkey, Bülent Ersoy is a major celebrity.

I'm not praising Turkey, I'd see Erdoğan gone if I could, I'm just getting some facts out b4 Woids say Turkey is Muslim therefore intolerant. It has freedom of religion too.

Also worth mentioning that Ahmad Yusuf is one politician among many, expressing one possible vision for the future of Palestine. There are other political currents swirling internally.

Big part of Palestinians are Christian and their doing fine. Homosexuality is currently not legal and we don't know if independent Palestine would have fixed it on not because Israel forbids them from holding elections. Same applies to most other issues, but one thing is for sure that they will have better agriculture and trade and more opportunities as a free country. Israel is essentially blockading Gaza and West Bank gets double tariffs and no water for farming.

EDIT: I’m interested in how a free Palestine would play out in terms of what that would mean for women, gays, children, people of non-muslim religions, in terms of personal freedoms, etc.

The basis of this question is Zionist because it implies things would get worse for gay Palestinians, Palestinian women, and non-Muslim Palestinians. All of these populations are currently either under military occupation or in exile, banned from entering any of historic Palestine. They may experience oppression or marginalisation based on these identities, but all of them additionally are under Zionism's boot as Palestinians. When they are no longer oppressed for being Palestinian, it will be much easier for them to seek liberation along the other lines they are oppressed, such as gender, sexuality, and religion.

As for religion, some Palestinian parties are specifically Islamic, some are explicitly secular. It would be hard to say, but right now we are seeing very strong unity between the different Palestinian factions, including e.g. explicitly Islamic factions strongly condemning the attacks on Palestinian churches. Hamas, an Islamic party, is in alliance with the PFLP, which was founded by a Palestinian Christian and is explicitly secular. From what I can see, religion doesn't seem to be a hugely dividing line among the Palestinian factions, and I hope that will continue to be the case when Palestinians begin the process of recovering from Zionism and establishing a long-term state.

In any case, Hamas, one of the most influential explicitly Islamic Palestinian factions, is also explicitly supportive of religious tolerance in its charter, and envisions a Palestinian state in which people are free to practise their own religions. I am hopeful that the current religious unity we can see will continue after the fall of Israel.