1) If "О" is a part of an unstressed syllable then it's reduced to something resembling "А". This phenomenon is called "vowel reduction". It's the complex and elaborate topic. You can check it out here on wikipedia. Vowel reduction in Russian
2) "Е" and "Э" are two different sounds, so it's mostly about how the word sounds.
3) And again, it's mostly about how the word sounds. The "Ь" is the symbol showing that the preceding consonant is soft, hence the symbol name, which is the soft sign
Although, you use the soft sign only if there is no other evidence that the consonant is soft. So let's see what can be the evidence of that.
If a consonant is followed by these vowels "я, е, ё, и, ю" then the consonant is soft. For example - неделя - "н, д, л" are soft in the word. However, sometimes we still have to add "Ь" before a consonant even though it's followed by the "soft vowel".
For example:
объятья and объятя would sound different for a Russian, the first one would sound properly and mean "hugs", the second one would sound improperly and be a nonsense word, however people, I think would still understand what you want to say with объятя but they would have to guess in this case
So the soft sign is usually put into consonants' clusters as in маленький
or at the end of a word as in тень
Note that the Russian soft and hard consonants sound completely different from each other to a Russian ear.
For example:
"мать", which is "mother", if pronounced without softening the final "Т" turns into the word "мат" and this word means "obscene language, swearing".