я, ю, е letters are pairs for а, у, э, and, мае should be read maje, and майе should be read majje, like it happens with мая and майя - maja and majja.
I, personally, think that э is a different vowel than е in modern Russian, unlike vowels in я-а, ю-у pairs, ie I think мяч and мач have same а, but only different м, but in case of мер and мэр I think the vowels are different. I do not know how old is this situation, but according to general logic that appear, е is just the match for э like я is for а, seems it was so historically. I do not ask about this, this is just to show the problem. I, personally, used to wrongly think that айе should be read aje just like ае, because I thought that е generally behaves differently from я and ю... For example, I thought, it does not change the m in меч, does not palatalize it, like it happens in мяч, so I thought, е has initial j only in beginning of word, so, I thought, е's behaviour after й is just like after any other consonant, while я after й should use its initial j and make a double j.
Some letter sequences here are not meaningful words, they are just examples, the question is about how letter sequence should be read.
This question is to get a proof to fix a wiktionary function/table, (I do not know what is it and how named), for https://en.wiktionary.org/wiki/User_talk:Benwing2#reading_of_%D0%B0%D0%B9%D0%B5_etc , for table in https://en.wiktionary.org/wiki/%D0%BC%D0%B0%D0%B9%D1%8F and other similar words.
Adding on December 18:
Some answers are about what happens in real speech. I think there is some current canonical Russian writing system, where should be a simple general rule, reading (pronouncing) according to which is not incorrect, it is correct, acceptable because it is canonical, generally it maybe a little archaic, not like people really speak, but ok. For example, pronouncing "gripəm" (or maybe there should be other thing instead of ə, i am not sure, it is not important for this question) maybe what is used by more people and more often, but if somebody says "grippəm" or "grippom" it is not incorrect, because it is somewhat canonical. I think using such canonical version would be more correct to use in a table in Wiktionary. I asked about such "canonical Russian writing system", not about current etymological/phonetical processes / phases of processes / changes "in the wild" (sound shifts etc). Also some little known pronouncing rules maybe not known, not noticed by some people, and may be pronounced by them without that rule, as it is written, or there maybe dialectal differences.
Adding on January 10, 2019:
Should "йе" after vowel be pronounced [jje]? Should [jje] after vowel be written "йе" or "ййэ" or "ййе"? Should [je] after vowel be written "е" or "йэ" or "йе"?