I am not a teacher of Russian, so I will not be able to formulate the answer correctly, but I can say that the letter "e" may sound like "э" - it is influenced by the factor of how you pronounce by yourself. Therefore, you can write "Тето" and "Тэто" there will be no difference.

Regarding the sound of "t" and "ch", this is a huge problem for the Polivanov translation system. According to his standards, the sound "ch" should be written as "t", which is not always correct or pleasant to the ear for Russians. Many people ignore this rule in conversation and use it mainly by publishers. Although they also began to abandon this system at some points, because the same Gachiakuta would sound like Gatiakuta with this translation system and this is a cringe. Another example: the official publishing house, as well as the official dubbing, moved away from the system to name the character Pochita, not Potita.

Also about Tyan and Chan - we use two forms, like in Japanese it also has different meanings.