It's not מאני because only the object pronouns attach to prepositions like ב or מ while אני is a subject pronoun. As for why it's not מני, honestly I don't know lol. But to the best of my knowledge it's just one of those things in language where there's no rhyme or reason to it and you just have to memorize it.
That's correct, in fact there are certain cases in which מן by itself is still used (I'm not sure what the grammatical rule for it is, though).
For example, there's a Jewish blessing that goes "ברוך אתה ה' אלוהינו מלך העולם המוציא לחם מן הארץ" - blessed are you, Lord our God, King of the Universe, the one who brings out bread from the earth.
The vowel after the letter י in ילדים is, in Hebrew grammar, called a shva. When the shva comes after the first letter of a word, it is pronounced as a brief "e" sound.
But adding the ה which makes the word הילדים means that the shva no longer comes after the first letter of the word. It instead is silent, and it indicates that the י is the last sound in the first syllable. Thus ילדים is pronounced "ye-la-deem" but הילדים is pronounced "hay-la-deem".
The shva itself looks like this: ְ (it's the two reeeally tiny dots!). With full vowel points the words ילדים and הילדים look like יְלָדִים and הַיְלָדִים
As for whether a native Modern Hebrew speaker makes that distinction in day-to-day speech, I don't know.
Hope that helps :)