The first letter of the Arabic "al-" ("ال") prefix is known as a hamzat al-wasl. This can be written either as an alif with a waslah above it "ٱ", or just as an alif by itself "ا" (e.g. with no hamzah).
While a regular alif (with a hamza) will always be pronounced with its own vowel as a glottal stop, a hamzat al-wasl is silent when it is preceded by a vowel (and if it's preceded by the prefix "li-" ("ل"), the hamzat al-wasl is dropped entirely; such is the case for "لله" in Al-Fatihah 2). If, however, it's at the beginning of speech you would pronounce it as a regular alifat the beginning of speech you would pronounce it as a regular alif.
That's why you will typically hear (and recite) "Al-" in full at the beginning of, for example, Al-Fatihah 2 & 3, but only hear (and recite) "-l-" elsewhere.
This is not limited to the "al-" prefix; another example of the hamzat al-wasl would be the first letter of "اهدنا" in Al-Fatihah 6 (but since it's at the beginning of an ayah you'd probably be pronouncing it anyway).
As for the final vowels, they're just not pronounced at the end of speech; it is common practice to stop between each ayah, so the final vowels would normally be ignored. If, however, one chose to recite multiple ayat together as a single recitation, these final vowels would be pronounced (and possibly absorbed into subsequent hamzat al-wasl)
So, using Al-Fatihah 2-3 as an example:
Ayat recited separately
2 Alhamdu li-llahi rabbi-l'alamin
3 Alrahmani-lrahimAyat recited together
2 Alhamdu li-llahi rabbi-l'alamina 3 -lrahmani-lrahim