I’m not an expert in middle eastern politics.

My uninformed take is that if the Gaza Strip was to self organize as a state, they will implicitly recognize Israel as a state as they will have to decide on borders.

Being unrecognized allows the Palestinian authority to claim all the former Palestinian land as their own.

This is already the defacto situation for palestine. It is already recognized by many countries. Treaties have been signed. We've been close to a two state solution in 2007 when PLO was in power. Hamas however is not about being at peace with the situation and aims at the dismantling of Israel. It wants an Islamic state from the east to the west. At multiple occasions has it sabotaged a two state solution through violence. Hamas is everything but stable, it is an Islamic extremist faction. The threat of hamas/isis has put religious extremist nationalist factions into power in Israel also fueled by people with a religious conviction. Just that the land of the palestinians belongs to the jews. They are not particularly interested in peaceful coexistence as seperate countries either.

I think most people have given up hope of a two state solution. It looks further away than ever.

https://www.un.org/unispal/history/

Nothing would change. For a start, PA doesn't control gaza so can't service it, and secondly israel and Egypt are both blockading Gaza. Independent Gaza would continue to have nothing.

Palestine, like Taiwan, has a non-voting representative in the UN. It's effectively its own country save for the key players of the Security Council recognizing them.

The Gaza strip is effectively ruled by Hamas, which rule without election since 2006. If they're to declare independence, aren't they need to elect a legitimate government first?

They can declare whatever they want, but if the major nations of the world do not recognize their declaration as valid and support them then it's a hollow victory. The PLO declared an independent Palestinian state back in 1988. Many nations recognize it as an independent state today, and many nations do not.

https://en.wikipedia.org/wiki/International_recognition_of_the_State_of_Palestine

A "Gaza country" would lie between Saint Vincent and the Grenadines, and Grenada the list list of countries by size (place 185). and between Lithuania, and Qatar in the list of countries by population size (Place 140). With no agriculture to speak of. How could such a county exist?

Why do you think Hamas would be interested in peace with Israel? They very explicitly have never been and are not.

"it would likely have a stable government in Hamas"

....re-read that to yourself a few times.