"Alle = all" can only be used with plural nouns (e.g. alle Kinder = all children). There are different forms of "alle" depending on the case of the noun: nominative/accusative: alle, dative: allen, genitive: aller.
"Jeder, jede, jedes = every/each" can only be used with singular nouns (e.g. jedes Kind = every child). Which form you have to use depends on the gender and case of the noun, similar to the different forms of the definite article. You can see all the possible forms here (first table on the right): http://de.wiktionary.org/wiki/jeder
Yes, mjondras, you are correct. 'Every child likes her.' - 'Jedes Kind mag sie.' 'She likes every child.' - 'Sie mag jedes Kind.' But be aware that the sentences look so alike because the verb is conjugated the same for "es" (das Kind) and "sie". If one of them were plural: 'She likes all the children.' - 'Sie mag alle Kinder.' 'All the children like her.' - 'Alle Kinder mögen sie.'
"Sie" as in "they" would use the verb without conjugating it (i.e. haben, spielen, etc.) ; "sie" as in "she" uses a conjugated verb (i.e. hat, spiele, etc.) If you research conjugating German verbs, there's a pretty general structure to how a verb is used in relation to the noun (ich habe, du hast, sie/es/er hat, Sie haben, etc.).
That would be logical, but unfortunately, like the other modal verbs "mögen" is irregular. In the present tense singular, there is a vowel change and the first and third persons singular (I + he/she/it) get no endings.
mögen (to like)
ich mag (I like)
du magst (you [singular informal] like)
er/sie/es mag (he/she/it likes)
wir mögen (we like)
ihr mögt (you [plural informal] like)
sie/Sie mögen (they/you [formal] like)