How sound is the idea of Allah extinguishing the Fire and the disbelievers emerging from it after a period, and is dwelling in Paradise likewise limited? And what is the ruling on the idea of limited eternity and unending perpetuity?

There is no hadith stating that Allah extinguishes the Fire by placing His foot in it. Rather, what is authentically reported in the Sahihayn is that the Fire will not be filled until the Lord of Might places His foot in it, whereupon it will say: "Qat qat," meaning, "Enough, enough." This indicates its filling, not its extinguishment or annihilation.