In words such as
I have looked through dictionaries and cannot find why in these words the «и» at the end sounds like the letter e=ye. Why is that, what is the rule?
Sounds of a language are a fuzzy concept and when it comes to mapping them to letters, a lot depends on the listener's background. Many things only exist in the eye of the beholder (or the ear of the listener).
Many Russian speakers heard the English refrain "She's got it" as "Shizgara" («Шизгара») with the original 't' mapped to 'r' and the weak 'it' to weak 'a'.
I suggest that you just assume that there is an И at the end of these words and continue to practise listening until you 'get it'.