Is it permissible to recite during prayer with some Tajwid due to not being familiar with all of its rules? And what is the ruling on lengthening the word "Amin" after Al-Fatiha, and is this lengthening permissible in normal recitation or with some Tajwid?

The obligatory aspects of Tajwid, both within and outside of prayer, involve pronouncing letters from their correct articulation points, not substituting one letter for another, and avoiding errors (laḥn) that alter the meaning. Neglecting these aspects is impermissible. As for other elements aimed at beautifying recitation, such as observing the madd (prolongations), these are recommended, and prayer is not invalidated by their omission.

One should not abandon known Tajwid rules; rather, one should implement what is possible of them. It is preferable to learn all the rules and apply them to enhance the voice when reciting the Quran.

As for the prolongation of the alif in "Ameen" after Al-Fatiha, it is prolonged and lightened in all narrations and recitations. Several linguistic forms of "Ameen" have been reported, the most famous being prolongation and shortening, with the mim being lightened in both cases. The most eloquent of these is prolongation with lightening.

It is permissible to prolong the alif in "Ameen" after Al-Fatiha, just as it is permissible to shorten it. The prolongation in "Ameen" includes the natural madd (two counts), the moderate madd (four counts), and the saturated madd (six counts). Thus, it is like Madd al-Badal, where shortening, moderation, and saturation are permissible. Similarly, the prolongation of the ya’ in "Ameen" when pausing (Madd al-Aarid li al-Sukoon) allows for the three facets: shortening, moderation, and saturation, whether in prayer or in recitation outside of it.