Is pronouncing "Lord of the Worlds" as "Rabbil 'alamin" and pronouncing "the Most Merciful" and "the Straight" without the "al-" (definite article) correct, and if I made a mistake in Al-Fatiha, what is the ruling on previous prayers?

"Rabb al-Aalameen" (Lord of the Worlds) is read with a shaddah over the baa, and the lam in "al-Aalameen" is lunar (qamariyyah), and one does not pause on it. As for "al-Rahman" (the Most Gracious), it is read with a solar (shamsiyyah) lam, and "al-Mustaqeem" (the Straight) with a lunar lam. Therefore, the way you read it is correct. You do not need to repeat your prayers, and we advise you to learn correct recitation.