If I do not do that roll/trill of the 'r' sound in the Takbeer (saying Allahu Akbar [Allah is Most Great]) or Surat Al-Faatihah even though I can, is my prayer still valid?

All perfect praise be to Allah, the Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.

Your prayer is valid if you do not repeat the ‘r’ in Takbeerat Al-Ihraam (opening Takbeer), because that is the correct way of pronouncing it (without repetition). Repeating the ‘r’ (i.e. pronouncing it with Shaddah) is a mistake that renders the prayer invalid according to some scholars of Fiqh, and it is not failing to repeat it that renders the prayer invalid, as you fear. [The Shaddah is a diacritic which replaces a double-consonant; which is equivalent to writing the same letter twice]

Pronouncing the ‘r’ with a trill (with Shaddah) is adding one letter. Ar-Ramli mentioned the difference of opinion among the scholars of Fiqh of the Shaafi’i School in this regard in his book Nihaayat-ul-Muhtaaj, and he favored the view that this does not render the prayer invalid. He said:

“Adding one letter that changes the meaning is harmful, such as prolonging the Hamzah (the first letter ‘A’) in the name ‘Allah’, and in pronouncing Akbar by adding the letter alif (أ) after the ‘b’ in Akbar, as it becomes Akbaar, plural of ‘kabar’ which means a drum with one face, and adding the letter ‘و’ (i.e. saying ‘wallahu Akbar’) before the name ‘Allah’ as in the Fataawa of Al-Qaffaal, and pronouncing the ‘b’ or ‘r’ with Shaddah in Akbar, as Ibn Razeen issued a fatwa to this effect, which is evident in the first part (i.e. in pronouncing ‘b’ with Shaddah), but as regards the second part, which is that pronouncing ‘r’ with Shaddah harms, then this is rejected, as stated by Ibn Al-‘Imaad and others, because the 'r' is a repeatable letter (naturally pronounced with slight repetition) so adding an ‘r’ (i.e. pronouncing it with Shaddah) does not change the meaning.”

The author of Mawaahib Al-Jaleel said, “Beware of prolonging the Hamzah (A) in the name Allah, as this may be understood as a question (آلله); and of prolonging ‘b’ in ‘Akbar’ thereby changing the meaning (to sound as أكبار); and of prolonging the Dhammah (oo sound) at the end of the name Allah (pronounced Allahu) to the extent of adding the letter ‘و’; or to pronounce the letter ‘r’ with Shaddah; all these are mistakes in pronunciation that are feared to invalidate the prayer.”

With regard to not repeating the ‘r’ in al-Faatihah, if by this, you mean leaving out the Shaddah (which is equivalent to writing the same letter twice), then the Hanafi School is of the view that this does not invalidate the prayer if it does not cause a change in the meaning. If it changes the meaning, then there is a difference of opinion about it invalidating the prayer.

The author of Al-Muheet Al-Burhaani said, “Leaving out the Madd (prolongation) and the Shaddah (a diacritic which replaces a double-consonant) in their proper place and articulating them in other than their place, if this does not change the meaning, then it does not necessitate invalidating the prayer, and if it changes the meaning, then the scholars differed in opinion about it; some of them said, ‘The prayer is not invalidated, as a way of warding off hardship,’ but most of them said, ‘The prayer is invalidated.’”

The Shaafi'i and the Hanbali Schools are of the view that the prayer becomes invalid by leaving out the Shaddah in Al-Faatihah. Al-Maawardi from the Shaafi’i School said in his book Al-Haawi, “As for pronouncing the verses of Al-Faatihah with Shaddah in prayer, then there are 14 places of Shaddah in it, and leaving out the Shaddah is not permissible because the letters that have Shaddah represent two letters, so if the Shaddah is left out, then it is as if one has left out one letter, and therefore it is impermissible.”

The author of Al-Mubdi’ (a Hanbali book) said, “If one leaves out a Shaddah in Al-Faatihah, he is obliged to repeat it, because the Shaddah substitutes a letter, and if someone leaves out a letter, it is as if he did not recite it, because a compound thing is non-existent if one of its parts is non-existent, but Al-Qaadhi mentioned in Al-Jaami’ that the prayer is not invalidated by leaving out a Shaddah.”

Hence, you should recite Al-Faatihah accurately while observing the Shaddahs that are in it, as long as you can do so, as you mentioned in the question. Pronouncing the words with Shaddah is easy and does not involve hardship, and you should beware of Waswasah (obsessive doubts) in this regards.

For more benefit, please refer to fatwa 350782.

Allah knows best.