Is it obligatory to observe the rules of Tajwid in prayer, even in places other than the recitation of the Qur'an, such as the opening takbir (Takbirat al-Ihram) and the Tashahhud?

There is no obligation to observe the rules of Tajwid outside of the Qur’an, though it is preferable to observe them without affectation. The takbir (pronouncement of "Allahu Akbar") in prayer includes elements that are pillars (like the takbirat al-ihram), and elements that are obligatory or sunnah. It must be recited correctly. Qalqalah occurs when a letter is sakin (vowel-less), and the letter ba in "Allahu Akbar" is not sakin. Similarly, the tashahhud includes elements that are pillars, obligatory, and emphasized sunnah. It must be recited correctly. The noon in "an la ilaha illa Allah" in the tashahhud should not be pronounced with izhar (clear pronunciation) because idgham (assimilation) is the standard recitation. However, pronouncing it with izhar does not invalidate the prayer because it is a minor linguistic error that does not distort the meaning.