Is it permissible to supplicate with, "O Allah, I ask You for Paradise and I ask You for Al-Hur al-'Ain, and I seek refuge in You from the Fire" after every prayer, and is this specific to the prescribed prayers?

Supplication (Du'a) is permissible at all times, especially after performing acts of obedience. It is sufficient to ask Allah for Paradise and to seek refuge with Him from Hellfire, and to strive to fulfill the means that lead to Paradise. If you enter Paradise, you will find therein whatever souls desire and eyes delight in, and bliss beyond your imagination—an everlasting, perpetual comfort. This is based on Allah's saying: "And therein is whatever souls desire and eyes delight in. And you will abide therein eternally," and His saying: "No soul knows what has been hidden for them of comfort for eyes as reward for what they used used to do." Whoever enters Paradise will attain therein more than they could ever wish for.